Can all 3 computers access the internet? If yes, then the problem will most likely be with limewire or the firewall/network settings on the problem computer.

Things to check/do:
1) Have you restarted the problem computer at least once before posting? Sometimes this is all it takes to fix the problem.

2) Make sure that the network settings on the problem computer are the same as the other two. Click on 'Start' --> 'Control Panel' --> double-click 'Network Connections' --> right-click on 'Local Area Connection' --> select 'Properties' from bottom of list that appears --> double-click 'Internet Protocol (TCP/IP)'. Make sure the settings are the same as the other computer that is running limewire.

3) Check Windows firewall settings. Maybe something got changed and you didn't realize it. Click on 'Start' --> 'Control Panel' --> double-click 'Windows Firewall' --> click on 'Exceptions' near the top of the box that appears. Is limewire listed?