A Wireless network card isn't going to do it to connect the Wii to the internet. You'd need to use the now discontinued USB 'dongle'. Quick question, how is your comptuer connecting to the internet if you don't have some type of wireless router? You should be able to use the wireless function on the Wii to connect the same way you connect your computer as long as your wireless router doesn't rely on WPA security (a Wii can only use WEP) and the router is a compatable type.
Good luck and if you give us a little more info we'd be able to give you some more hints and help on how to get on-line.