Have you tried manually appling network setting on both computers? Try setting the desktop IP to 10.0.0.1 with subnet mask 255.0.0.0 and the laptops IP address to 10.0.0.2 with the same subnet mask. Both Default gateways to 10.0.0.1 and see if it works! I may be wrong in the actual address and reckon if so, someone will correct this so it is the right numbers but you can probably see where im going with this.
Or as you are using a modem and not a router in your first port, you could use the 192.168.1.1 with subnet of 255.255.255.0!
I maybe wrong but sounds like it could work!
In regard to network services that are set to OFF, Network diags should re-enable them back to how they should be!