I'm fairly sure I didn't do anything wrong, the first or fifth time, but I actually needed to get the updates in a specific order. I got around the problem by downloading Service Packs 2 and 3 to my machine. After starting with a new install of XP, I installed SP2, restarted. Then I got the latest version of BITS, IE7 and Genuine Advantage via the MS website, and installed them. Then I installed SP3. After all that windows update still worked and I was able to get more updates via the normal update interface. It was all very frustrating trying to guess how to get around some problem in the upgrade path.
The happy news is that I have a new machine that runs really quiet and fast, and a big new flat-screen monitor.