I started to dismantle the rack. At this stage, I can share one tip: loosen both the inner and outer tie-rod ends before you remove the rack from the car. Tomorrow, I will start the actual rebuild.
Otto, take some pictures if you can.
Otto …
8 out of 10 … based on your apparent online knowledge and level of commitment … I think that bet is a WINNER 
P.S. Your pictures have been excellent 
Today, I had difficulties when removing a collar at the passenger end of the rack. Finally, I figured out how to remove it. The black square aluminum mount can be held in a vice and then you can use a hook wrench to unscrew the collar.
I thought that this teflon seal is broken, but it is not. The new one is a band which will be wrapped in place.
I have now renewed all the seals and assembled the rack. I will provide pictures later.
I am still a bit confused about the lubrication of the rack and pinion, namely should I grease the toothed section of the rack. 
I think the toothed section is not lubricated by power steering fluid, because if it was then fluid would leak to the driver’s side inner tie-rod. Probably I will add some bearing grease to the teeth: https://www.ez-catalog.nl/Asset/f3005d378bc041c598248747f50cef4e/Original/SKF-grease-EN.pdf
What do you guys think?
Otto …
you asked if you should grease the tooth rack …
This guy did. Although it’s not an XJ40 rack the basics are the same.
Dennis,
Thanks for your support. I added grease to the teeth. I also painted the rack housing and assembled the rack back to the car. The left tie-rod went almost to the steering knuckle, but the right side was 20 mm off! The steering shaft-to-steering gear pinion shaft U-joint was correct since I had market it. The tie-rod end distances were correct, too. So, I think the only option is that the pinion is not on the correct spot or tooth" of the rack. I will take the pinion off and try to move the rack to the right and put the pinion back again. However, I find it strange that the left side was almost aligned.
Way back in my memory, and I could be wrong, but I think there was a hole somewhere that you put a pin in to centralize the rack in the housing.
Otto …
No need to take the rack apart.
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Just disconnect the universal fitting that connects the steering shaft to the power steering unit.
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Using some pliers or vice grips rotate the power steering shaft from lock to lock and measure those two distances from the tie rod to the power steering shaft on one side.
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Divide that distance in half and rotate the shaft so that “half” distance is now between the tie rod and shaft. You have now centered the rack.
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Now center up your steering wheel and reattach the steering shaft to the rack via the universal joint.
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The rack is now centered but of course depending on where your tie rods were previously adjusted the toe on the wheels may not be aligned properly.
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The earlier XJ6 models did have a hole that you could stick the smooth end of a drill bit in to center the rack. I don’t believe the XJ40 racks do.
my mechanic changed the pinion seals with the complete kit but the oil leak has not disappeared … would like to try to fix it by building a bushing … could it be the pressure of the power steering pump? maybe it just needs to be calibrated or replaced …
Max …
I believe I would completely discount the idea that your power steering pump is causing your power steering rack to leak. There is no “calibration”, it’s simply a vane type pump that runs off a gearing on the engine.
Power steering racks (especially ones that are constantly being rebuilt during their 3 decades of life) simply wear out. The “O” rings and bushings take their toil on the metal parts they slide over and even when they are rebuilt with new seals they can start to leak again before long… Does anyone know of any company that is making ALL new racks and not just replacing the seals and reusing the old metal parts?
I recently went through 3 (count 'um 3) racks from a very well known and reputable company before I finally got one that didn’t leak after using it for a short while. So make sure you purchase one from a good company that offers a lifetime warranty (there’s a better than even chance you’ll need it).
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I have heard from a former Jaguar service manager that he had good luck with rebuilding them IF he had the seal wearing surface professionally polished before reassembly.
Dennis,
Thanks for your description of centralising the rack in the housing. I actually took the rack apart before I got your description 
. I tried to adjust the rack as well as possible so that it would be centralised in the housing. When I put the rack back to the car everything was in place! I bled the system and moved the car a bit in my yard. So far I have not noticed any leaks. Let us see how it goes in the long run.
Overall, I would say it is not very difficult to rebuild the rack by replacing the seals, but it takes time. I am happy to share my knowledge if someone is willing to replace the seals of his rack. Dennis, thanks for your contribution and encouragement. You do not need special tools, except tie-rod tool for tie-rod outer end removal and hook wrench for rack collar removal. I used this cheap (8 $) wrench: https://www.biltema.fi/autoilu—mp/moottoripyoraily/korjaamovarusteet/mp-tyokalut/hakasavain-iskunvaimentimelle-2000037357
I painted the rack with silver Hammerite (still wet in the picture 
) before I put it back to the car.
I performed a leak-down test. The leak percentages were 8-13 % which are appropriate for a 32 years old engine. The leak-down test, cylinder compression test (170 psi for all cylinders) and vacuum tests I have done indicate that the engine is mechanically sound. For the leak-down test, it is crucial to set a top dead center for each cylinder precisely.
I have also been cleaning the engine bay around the battery area and painted the battery tray with black Miracle Paint. Luckily, the bulk head of my car is in good condition.
Next, I will add Forte Power Steering Treatment (125 ml) to the power steering reservoir as suggested by a local independent Jaguar repair shop.
Otto …
Great news on the engine tests. Now you can eliminate the internals of the engine in future troubleshooting, and that usually narrows things down quite a bit.
Thanks for the excellent pictures of your steering rack rebuild (you do take great pictures). Hopefully that will solve your leaking problems. And if it does you’ll be an inspiration us all. Not many people (myself included) have gone that route.
Dennis,
I do have more pictures of the steering rack job. However, the pictures illustrate the various seals of the rack. These seals are quite obvious once you have taken apart the rack.
Concerning the engine tests and all the parts I have either replaced or tested, there should be only two options left for the little rough idle. These are the fuel pressure regulator and the ECU. As I have reported earlier the fuel pressure test revealed that fuel pressure did not decrease once I had reconnected the vacuum hose to the regulator. I should get a new regulator tomorrow. Let us see if a defective regulator is the cause for the rough idle.
Finally, I received the Delphi TSP0585012 thermal expansion valve. I replaced the original valve immeaditely, and yes, the Delphi has correct threads. 
I hope I get the fuel pressure regulator tomorrow.
I replaced the fuel pressure regulator. It is easy to change. It is not necessary to "remove the fuel rail and the injectors as
an assembly" as instructed in the Haynes manual. However, the new regulator did not improve the idle. When the engine is cold there is some hunting. So, I am thinking that probably the stepper motor is defective.
According to Haynes manual, page 6-6: "Check for power to the MAF sensor.
Backprobe the MAF sensor electrical plug.
Working on the harness side with the ignition
ON (engine not running), check for battery
voltage on terminal number 5 (see illustration).
40 Remove the pin and backprobe the MAF
sensor electrical connector terminal number 3
with the voltmeter (see illustration). The
voltage should be less than 1.0 volt with the
ignition switch ON (engine not running). Raise
the engine rpm. The signal voltage from the
MAF sensor should increase to about 2.0 volts."
To my understanding, the MAF voltage can be adjusted by turning the allen screw in the recess at the side/corner of the MAF unit. I have read from Internet sources that 1.2-1.4 volts would be ideal. Has someone adjusted the voltage by turning the allen screw?
Overall, for some reason I do not get any voltages when I backprobe the MAF as instructed. However, I do not believe that my MAF is totally dysfunctional since that would cause clear symptoms most likely. Strangely enough, I have proper resistance across the air temperature sensor terminals. Additionally, according to this source, MY 1988 should have voltage only at idle: https://www.jaguarforums.com/forum/attachments/xj40-25/10164d1304884477-88-vanden-plas-still-wont-start-engine_management-xj6-vdp_1988my-pdf
Here is the Haynes manual (scroll to page 6-6): https://www.jaguarclub.com/wp-content/uploads/2015/08/XJ6-Service-And-Repair-Manual-Haynes1.pdf
Here is a description of MAF: https://www.jaguarforums.com/forum/attachments/xj40-xj81-25/10097d1304573388-testing-xj40s-air-flow-meter-testing_xj40-airflow_meter.pdf
Otto …
Well since you’ve determined that the internals of your engine are good (and you can take a big sigh of relief for that) that only leaves three things that can be the cause of your rough idle …
Vacuum leak
Erratic fuel delivery
Electrical fault
The vacuum leak should be the easiest to troubleshoot. All the vacuum lines originate at the intake manifold. You can remove each line one at a time and cap the nipple at the manifold. If one results in smoothing out your idle you’ve discovered that there is a vacuum leak somewhere downstream on that line. Next you can do a test with a quality vacuum gauge and finally do a smoke test.
If you have sufficient pressure in the fuel rail then you’re down to the injectors. Either being fouled or not receiving the correct electrical pulse from the ECU. There are test to check both of these problems.
Finally you’re left with an ignition problem. Which start at the coil and flows through the rotor, cap, and spark plugs, along with the associated wires. If the car runs fine at higher rpm’s I would discount the ECU as the cause.