Of course, it turns out that I couldn't get the license number for Sequel from the copy-protection License Control Center, because it happily GPFd when I tried to run it. I've been on the phone again with Steinberg support which referred me to Syncrosoft support which hasn't gotten back to me since I e-mailed them this morning.
Adding to the general entertainment of the situation, it looks like one of the two CPUs in the computer sometimes fails to work, at least if the Windows Task Manager can be believed. That's not good either. *sigh*
I'd really like this computer to keep working, because the taxes aren't done yet.