For the second pc have you tried configuring it yourself with a static IP?
Manually configure the adapter to use the same gateway address, subnet mask and DNS servers.
For the IP address setting use an address thats falls into the same subnet. For example if your gateway address is 192.168.0.1 I would set it so that my static IP address was say 192.168.0.10 or 192.168.0.11 the last octet can go up to 254. You only need to change the last octet (in bold). Leave the first three octets the same as the gateway.
See if you can get internet connectivity then. If you still can't get on the internet try pinging your cable modem (gateway address) and let us know if you get responses.
[EDIT] If this fails also try bypassing the switch with the second pc and connect it directly to the modem... If you get connectivity then, the problem is at the switch.
If you don't get connectivity try pinging the network adapter (127.0.0.1) if you don't get any replies the problem there.[/EDIT]