1. go into mint 10 laptop, open terminal and type ping -c 3 127.0.0.1. post results
2. go into both computers, one with the wifi connection and one with the ethernet connection, open terminal and type ifconfig post it here.
3. go into your linux mint and find "Network Connections" (should be somewhere under System -> Preferences ), or right click on the network icon in your system tray and click edit (i think). Go into Wired, select your eth0 (should be there) and press "Edit..." on the right. Tell me if in the "IPv4 Settings" tab "Method " is specified as Automatic(DHCP) or not?
4. try to connect directly to router with ethernet cable.