Instead of automatic detection, you may need to manually enter your DNS settings into the Wii console in order to connect to the Internet. To determine these setttings, you'll need to follow the steps below using your PC.
Another device may be sharing the same I.P. address of the Wii console. Try manually assigning an IP address to your Wii console. Use XXX.XXX.XXX.X+1 where X is the IP address of your PC. (or +2 if you have two PCs on your network)
To get your IP address and DNS settings:
Click the "Start" button in the lower left-hand corner of your computer.
Type "cmd" into the box and press Enter.
Type "IPCONFIG/ALL" and press Enter.
Enter in the exact same values for Primary DNS and Secondary DNS into the Wii console.