... So I believe the reason it won't work is because the server is binding to the localhost IP, and not the computer's IP address?
No, if netstat -an shows that your server is LISTENING at 0.0.0.0, then you should be OK.

Please review this thread and post the other information requested. I still don't understand your reluctance to show actual IPs. They should be private LAN IP's which are not routable, so there would never be a security issue. If they are actually public, routable IPs, then that's your problem, since the client can't find the server on the local LAN.

Again, being ping'able is good, but it doesn't guarantee that both machines are on the same LAN.

Does your server have two NICs?