Can't say exactly what is happening, but sounds like your Wii is not getting an IP address via DHCP. I doubt this will help much but you might want to look at this page:
I would also suggest:
a) making sure your PC works fine (you might want to get output of "ipconfig /a" for setup info on your wii)
b) double check wii's DHCP settings
c) talk with the admin's at school to see if they can help
Would like to help more, but my Wii is still waiting for me under the Christmas tree.