After messing around with the main batteries, the backup battery, the DC jack, the contrast knob and the reset hole a bunch, it’s definitely not working right. Twice, seemingly randomly, I did manage to get it to light up with the boot screen and chime, but it went off again in a second.
On closer inspection, there are signs of corrosion on the battery terminals! But it doesn’t seem to have gone far - nothing is visible on the dock terminals, the serial port or DC jack.
Let’s open it up!