It turns out the solution was adding the IPX protocol. I tried this after doing a google search:
Google
And the search result on
www.annoyances.org gave all kinds of suggestions. Some of these worked for some people, and not for others. I tried suggestion after suggestion until I added IPX. Adding the IPX protocol must make some configuration change that enables something that I don't understand. I didn't add IPX on the win2K box. I only added it on the XP box where I was having trouble. So I wasn't really using IPX, but just having it installed allowed windows to talk to windows. It doesn't make sense. It is like me learning German so I can speak English to someone.
I hope noone else ever has to deal with this one.