My reward for clicking the "test" button in the APC UPS software was that it completely cut the power to my SQL server. The server promptly rebooted, the SQL service is running (verified by the app on another server that queries it), but the GUI has remained on "applying computer settings" for 20 minutes and counting, and it forbids RDC connections. Since SQL is up, it is fulfilling its key role, but it's obviously not right.
What step to take next? Wait longer?
Hardware is a Dell Poweredge 2850, internal RAID10.
Personally...I'd test to see if I can remotely access the \servername\c$ share, and see if it's responding. Next see if there's any activity in the Users (Documents and settings?) folder to see if the profile is even doing anything or accessing anything, and maybe try checking the logs from another machine to see if it's responding to those requests and may have a hint of what's happening.
Then I'd use one of the pstools to see if I can get the system to shutdown (forced shutdown if necessary). reboot with a repair console or repair disk, and check the drive for errors with chkdsk.
If it does restart to a full state, see if you can remotely browse the C$ share again, and I'd try renaming the profile of the user you're trying to log in as, so it has to recreate it (in case the profile is corrupt).
It sounds like something may have been corrupted. If the database is running and there aren't any SERVICE FAILED TO START errors on the console, it may very well be the user you're trying to get in to the system as.
In case this helps someone else someday...
Given enough time (somewhere over 90 minutes, I couldn't supervise it completely), it did allow a local console login. Once I was able to taskkill the APC pbe agent, everything else started working properly.
Following that test, something happened to the APC powerchute software, and its failure to load caused numerous other problems.