I plan to replace that dead Sony LiPo cell shown in my previous post above with something cheap, like this:
3.7V 1200mAh 25C Lipo Battery $3.23 (Free Shipping)

http://www.ebay.com/itm/like/371474331938
Hopefully, the little circuit board inside the battery module will identify it as an authentic kindle battery, so the system will boot up. We shall see, won't we?
I have a K2 in this condition (worse actually -- the swelled battery split the side of the case open, breaking some of the tabs that snap the cover together). I will do "battery surgery" on it too...
If that works, I may try a somewhat physically larger (higher capacity) LiPo, but it needs to fit in the battery frame. A tiny battery may need recharging more often, but still probably less often than my cellphone...