I managed to get a document into a “stuck” state. The printer showed the document in the list, and it would show it as paused, as printing, as deleting. But it would never do anything. The solution? Restart the print spooler service. Worked like a champ.
Of course, now I have another document hung up, but I know how to deal with it. And then I can try to figure out the problem with this thing.