Well, good news and bad news.
The good news is that leaving it to charge over the weekend fixed the appearing-and-disappearing-on-the-USB-bus issue; the battery must have been dead flat. It shows up as a 15a2:0052 "Freescale Semiconductor, Inc. i.MX50 SystemOnChip in RecoveryMode".
The other good news is that my bodged together needles-masking-tape-and-white-tack serial interface looks like it might work. I have some awful photos here.
https://goo.gl/photos/imx1GnfZxqTAHJWn8
(The probes are sewing needles pushed into the sockets of a PL2303 serial converter. I place masking tape over one of the grounding cans so as to avoid shorting out the needles, rest the needles on the edge, push them as hard into the solder pads as I can manage, and then bond them into place with white tack. Also notice that I've cut away the corner of the orange plastic insulator which was covering the pads.)
The bad news is that I'm getting absolutely nothing out of the serial port whatsoever. I don't know whether this is due to my interface not making proper electrical connection, whether the Kindle simply isn't transmitting, or whether I've fried the Kindle's serial driver by connecting the wrong voltage of USB serial converter...
In exactly what circumstances can I expect to see serial output?
mfgtool works fine. It's a bit frustrating that I can happily download and run images on the device but can't see any output. (You know what would be awesome? A mfgtool image that contained a Linux kernel and initrd and which talked USB serial. Bet there isn't one.) However none of the available profiles seem to do anything when run. I get the impression that the diags profile should bring up a menu on the Kindle's screen, right?