I took the K2 battery module apart even more. I borrowed a LiPo battery module (single cell 3.7v) from a small quadcopter. I put it in the kindle and plugged the kindle into my computer -- nothing --- no USB device detected. Then I plugged it into a USB charger --- again nothing --- no kindle power LED, frozen "critical battery" eink display...
But after a few minutes on the charger, everything changed:
The quadcopter LiPo has its own internal "protection" circuit board (and it is too thick to put the K2 cover back on). But this is just a test, until I find a LiPo that fits better.
And into the kindle it goes! Notice (above) that the plastic battery module frame is flexing -- it may need internal support (like those aluminum sheets) to make good contact with the battery connector.
It's alive! Well, at least the power LED came on after it was plugged into a USB charger for a few minutes. It does not yet show up as a USB device when plugged into a PC.
And a few minutes more charging time, another attempt at seeing if a PC can see it as a USB device...Yes! Umm... Wait, what? It showed up as a "lab126 RNDIS" device. Hmm... More time on the charger -- now it shows up as a disk device, but with no partitions (probably just that 0MB "low battery" fake disk device you can see mentioned in the GPL source code).
I have noticed that the voltage across the LiPo battery terminals has been increasing, so the kindle IS charging it. That is a good sign. But I need to let it charge longer before I can test it in any useful way...
Progress has been made. Hopefully this K2 was not destroyed when the swelled battery bulged out the front and back of the kindle case...
EDIT: Just to show how SMALL that LiPo battery really is, the "GIANT" black plug you see at the bottom of that bottom photo above is a microUSB cable. And you know how much fun those are to get plugged in right-side-up in a dark room in the middle of the night (without damaging the jack), eh?