Could be a bad ethernet cable, so try another cable if you haven't already.
Other than temporarily losing your LAN connection, your computer shouldn't be affected by restarting the router, so try that as well. Your computer should recognize the LAN once the router is working. (Of course you may want to investigate that virus...not sure how you manage to get by with that situation.)
Another thing to check is if the computer will work on the same router connection as the XBox (with the XBox's ethernet cable if possible). If it does, then you've narrowed the problem down to the XBox. If you're sure all network settings on the XBox are correct, you may need to contact XBox customer service for a repair.
For 6 months my Xbox live on the 360 has worked perfectly via an ethernet cable directly into my netgear router, however, 2 weeks ago it stopped accepting it and when I test the connection (up to 10 times) it doesn't work, it doesn't even try to test the connection and suggests I plug the ethernet into the Xbox even though its already connected to both the console and router. What should I do- I can't restart the router cos its connected to a computer with a virus that won't allow it to start up.