Hmmm...FYI, if you buy a switch (very cheap these days) you'll be able to have the cable from the wall connect to the "uplink" port on the switch, and then a cable to the laptop and the desktop...making them both connect at the same time
But that's besides the point.
Can you ping google?
In
cmd type
ping www. google.com
if it finds the IP address (72.14.207.99) and sends a reply...then I'll be baffled
if it finds the ip address, and 'times out' there's a couple of things that could be causing this.
if it doesn't find the ip address...then there's a couple of things that coudl be causing this - most likely the network plug itself (on the laptop)