ETA: I left the 180c charging over night (battery in the compartment, adapter plugged in), and this morning the computer started up and the battery holds a charge. If I remove the power adapter, it continues to run on battery power until the battery discharges (not long, less than 1/2 an hour, but I don't expect much from these old machines... my roommate's 180c barely lasted an hour on battery power when it was brand new).

If, however, I keep the power adapter plugged in and remove the battery, it shuts down within seconds.

I don't know if this helps with diagnosing which parts need to be fixed first - the power adapter, or something on the board (the battery, at least, appears to be weak but functional).

Another issue, that I can now observe since the computer is staying powered up longer: when I start the computer up from cold, there is a band across the bottom of the screen, about two inches, where the display is blanked out. The rest of the display is fine (it isn't stretched or compressed or anything) but the bottom part of the display just isn't there. If I wait a few minutes, the full display appears and works normally. If I restart warm, the band will reappear briefly at startup but disappear by the time the desktop loads, and once again the display is fine after that.

Is this a capacitor issue or a display issue? I am not sure what is meant by the "tunneling" that people mentioned in other threads.