Ooooh, OK. I've never tried changing the 22222 to 80 in the WL software, so I don't know what that should do!
John's whitelist sounds very reasonable for the school, but there IS that non-standard port 8080 setting in your IE LAN set up to consider as well.
I'm starting to think maybe that your school's network is segmented with multiple routers behind one gateway and that might be how 22222 is blocked. Under this idea, your computer there would be on one router and the WLIP another.
Edit: You can browse to the weatherlink.com website without being blocked from the school's computer, right?