I finally got to the hardware store to pick up a putty knife for the Mac Mini disassembly. I took the case off, found the drive controller connector, and gave it a little push. Then I plugged the power back in, and it booted up and recognised both the hard drive and the CD drive! So I reassembled the case and thought I was done.

It turns out that I was too hasty. The usual grey apple screen with the spinner came up first, then it eventually went away to the bluish screen. But by that point the hard drive kept making retry noises and it never got to the login window. So I disassembled the case again, and removed the drive assembly according to this disassembly guide. I cleaned and reseated the drive connector, but this didn't improve the situation. I still don't get to a login window.

I think the next step is to pick up a replacement 2.5" IDE drive and swap the drive out. I'm not 100% certain that the problem is actually the drive, but I think it's worth a try. Unfortunately, the most recent backup I made of the Mini is perhaps a couple of months old.