|I would start with ipconfig at a cmd prompt.|
Is the dns, gateway, etc correct?
"but not its domain name -- not sure what the significance of that is"
It means you aren't getting to the isp's dns server.
Since your wired connection works fine why don't you try wired connect to see if it works fine with this problem pc. If so you know it has something to do with the wireless side of things.
Give a person a fish, they eat for a day. Suggest they internet search and they learn a skill for a lifetime.