OK, I've had the unit apart and the problem seems to be just that the battery has died. When the battery's no longer connected to the board the device appears to work perfectly so long as I power it off properly before disconnecting external power.
Does anyone know a source for replacement batteries for the Kobo Glo? It's marked "GN345361" and is apparently a 3.7V, 1200mAh part. I've
found a source, but those don't come with the small white connector that connects the battery to the unit's board.