Hey Mike, I took a few snaps of the bits involved, hopefully of some help to other late car owners! (do early cars have different setup?)
The LH rear handle had been a bit stiff for a while. Lately it was only unlatching at the very last movement of the handle so I knew it was overdue for an adjustment …of course I ignored it until “tomorrow”. …anyway, about a month or so ago it quit working. My intention this week was to pull the panel off and adjust the latch mechanism, but when I opened it up for a look, no amount of adjustment could fix it - the lug had snapped off the handle.
Replacement rear handle ready for installation with broken original at right.
Getting the panel off is straightforward enough. First, remove the arm rest. There’s 3 mounting screws recessed in the arm rest - the screw holes may have trim plugs in them; pry out to access the screws. Once out the armrest comes free,
Next, take off the inside door handle trim. There’s a screw under a small square plastic cover under the inner door handle. Most cars have these missing, but if it’s in place, remove it by putting a penknife tip or small screwdriver in the slot and lifting it out. Undo the screw underneath and the handle trim can be removed.
The wood trim has to come off to reveal the screws holding the upper panel, gently ease it out at the inner handle end and slide it forward. It should release. Now you can undo the two screws holding on the upper panel. when they are removed, the upper panel lifts away. You can now pry the lower panel away from the door, it will remain attached to the door switches, speaker and puddle light. Support the loose panel and unplug the three connections. The door panel will then be free. With all the panel components removed, you can access the outer door handle mounting screws …more by feel than by sight!
The rear door handles are interchangeable L to R, so it doesn’t matter what spare you pick up, either side will fit. The fronts however, are unique.
The handle is held in place by two 8mm (?) bolts. To get a wrench on these bolts you have to move the polyethylene weather sheet out of the way and feel your way up into the door. These two bolts are screwed into the back of the handle and clamp this bracket in place.
The two slots in the bracket are where the handle bolts attach. To remove the broken handle, take the forward bolt out completely and just slack off the rear bolt enough to pull and slide the bracket towards you, forward and off. The handle can then be removed from the outside of the car.
When you have the handle off, you’ll see the broken lug (usually) still attached to the actuating rod. You can unscrew the rod from the nylon attachment - or - if you are an un-droppable sort, 
remove the locking peg by pressing and sliding it out of the nylon adjusting block.
View inside door cavity, rod removed, nylon retaining adjuster and locking peg at right
When I replaced the handle, I noticed, even after adjusting the length of the rod (moving the rod threads up and down in the adjuster) that the door still seemed a bit stiff. I had the handle on and off several times before I had a close look at the rod. Seemed to me that the s-bend at the top should be more of a right-angle bend so I took a pliers and bent the attachment point to a less of a slope.
This may have been unnecessary, but I had a thought that the rod could have deformed when the adjustment went out and as the cast handle had no flexibility, the rod took the stress until it couldn’t bend any further and the stress transferred to the rigid handle lug and… kapow!
adjustment rod with modified attachment bend
Anyhow, whether you do a “bend” or not is up to you, but I would suggest using some lithium grease on the nylon bits before reassembly. After you re-attach and adjust the length of the rod, hold the outer handle in place and see how the mechanism works before bolting it all back up. The latch should release with minimal movement and force on the handle.
Once you are satisfied it all works properly, replace the clamp and bolt it up. One tip - leave the rear bolt in place and loose on the handle before attempting to clamp it up as it is impossible to start that rear bolt with the handle in place - ask me how I know 
Anyway, hope that helps clarify the process for the yet uninitiated. 