OK, I played a bit more.
I charged the battery with 400mA for a couple of hours, then disconnected it and left it overnight. This morning it still showed 4.15V.
I plugged it back in the kindle and the green LED lit up. It booted and went all the way to the 'home screen'! The battery indicator showed 'full' and everything looked OK. Then I picked it up to reassemble the back, and it froze.
My guess is that there's a micro-crack in a trace somewhere, maybe between the CPU and the memory?
I tried to plug it back in a PC, and it shows up as 'SE Blank RINGO' on USB, so I fired up ATK. It recognizes the device, but it won't program it: 'Failed to initial flash or Do not support this Flash'.
I think I've got a broken PCB, but a working battery and screen on my hands. It doesn't bother me overly, as I've got a Paperwhite that should arrive in a couple of weeks, and I've got enough books on paper to survive a minor siege, but it would be nice to have a back-up device.
If Geekmaster's offer of a broken-screen demo unit is still valid, I'd be interested! (If it's possible to make the demo units work, which I gather that it is...)