1. Firewall configuration instructions (click on link) to configure the XP firewall. Follow closely & it shouldn't go wrong. Else see these instructions & it might be easier: User Cant Connect - LimeWire

2. Make sure LW on the computer is not using same port as that of the laptop. Can check via LW's menu bar, Tools>Options>Advanced>Firewalls.

3. Cox are reknown as p2p shapers or blockers, but generally only apply such techniques to specific users who are VERY high bandwidth users. I doubt they are your problem. Particularly since you can use LW fine on your laptop.

4. I have doubts you are using Dial up. If you are. Then that is an issue in itself. Due to its lack of bandwidth, if one computer is connected to the net, then the other computer will be very slow browsing or anything else. If you are using a wireless device, then it will almost certainly have an internal NAT firewall. However, if it supports UPnP then you should hopefully be fine.

Do (1) & (2) first & see how you go & let us know what happens.