You dont need a wireless adapter. Just go to the wii menu and then into internet settings and search for your connection. After doing this go into medal of honor and click on multiplayer.
Your Wii has a built in WiFi chip. If you actually plugged in something, you did something wrong. I am thinking that you plugged in the Nintendo WiFi USB Adaptor, right? The thing in the link below.
Like said it easy, in the internet settings take the first option and find your wireless router, enter your security code if you have one and it should then go and update your Wii with the latest upgrade automatically. You know you are connected when it does that.