What to do when Windows Update stops working

Over the weekend I had to repair Windows XP SP2 on Myron's computer. Since that time I haven't been able to apply Windows updates. I can download them, either manually by going to Microsoft's website or by using Automatic Updates, but they won't actually install. After digging around for a while it's apparently a problem that happens when you repair Windows. It's prevalent enough that Microsoft created a Knowledgebase entry about it: Updates are not installed successfully from Windows Update, from Microsoft Update, or by using Automatic Updates after you repair a Windows XP installation.