Several things could be going on here, let's try a few different things.

1: Double click the Wireless Connection icon in your system tray (the area where your clock is), then go to the support tab and then click repair. Repair it and it will release and renew your IP address. Try connecting then and see if it works. If not...

Step 2... In Internet Explorer on the top pane, go to Tools, Internet Options, Connections. Make sure Never Dial a Connection is checked and then goto LAN Settings and make sure you have no check marks in anything here. Does it work now?

if not, let's go to step 3...

But before we do that, what type of router are you using and can you connect to 192.168.1.1 (copy this into your browser with no http, www etc. behind it... just the numbers) from the dead computer? (If you can connect, it should ask you for a password, the default password is most likely admin or administrator, with no username). Let me know from here.