Hi. I just put a burst of power into the battery that brought it up to 3.6v - this was actually enough to get the 'Ok' menu, and i could reset, and its all working perfectly now.
(its still not charging the battery very fast though - i may totally charge it outside the PDA but don't have the means here
Here's a simple li-ion charger using the bq2057 chip. See page 7:
The 2057 will charge either a 4.1 or 4.2 volt battery (yes, there is a difference). It probably determines which type of battery you have by reading the battery's thermisitor. I can't say for sure. But it's apparent that you must use the battery's thermisitor lead for this to work correctly.
The Rsns resistor (in the example this is a .2 ohm resistor) can be adjusted for the charging current that you want. .2 for 1A (and I presume) .1 for 2A.
And of course you still have to supply it with clean power. But that could even be a battery pack if you like.
Digikey sells the 2057 for $3.04 each.
The rest of the parts look cheap. Please be careful.
The way that I would approach the "repair" of your Z would be to bring all three leads out of the Z to your homebrew charger. This way you can leave the battery in while charging it and you won't have to crack the case to get into the internals. This does mean that you would have to build a 3 conductor cable, but so what?
And I would be really careful the first few times that I used the charger - like in the other room when you start it up. Be sure to read the app notes carefully, check your math (but it really does look pretty simple - and foolproof).