1990 Sovereign (XJ40)  Handbrake

Rats - I lied.  Everything else is not working okay!

For quite a long time when I got this car, the PO had been driving it
with the handbrake stuck partially on, and I unknowingly did the same
for a short while before I went overseas and put the vehicle in
storage.

Now on thoroughly going over everything I found that the thumb-button
to release the handbrake does not in fact do so, and I can only
release it completely by prying out the coggy thingee with a BFS to
lower the handle far enough to actually release the drums completely.

This mechanism is stuck down beside the seat, and a bit hard to get at
to see what's wrong.  I can see an aircraft style crowned nut holding
the handle on to a shaft, and below that there are 2(?) nuts fastening
the mechanism below to the frame, I guess.

There are also a couple of screws and some other flat metal bits I can
see, as well as the pawl I pried up to release the thing.

Is there a right way (and a wrong way?) to dismantle the brake handle
to fix whatever is wrong with the release mechanism?

Thanks for any insight.

- Tom