Now that the machine has a decent amount of RAM, it was fast enough for me to go to HP's download link directly. Guess what! The link won't work with an XP machine--needs 7 or better. I guess if you haven't upgraded your OS you don't deserve to be an HP customer?
But just for the hell of it, I went to control Panel and clicked Add New Hardware. The driver I needed was in the system somewhere, and Windows installed it for me.
Moral: try Add New Hardware first.