What you need to do is do the connection set-up (not manual) and do it as you would normally. It won't connect, but that's okay. If you have the Internet Channel installed on your Wii, go there and try to go to any page. If my understanding is correct, you should be redirected to a page where it asks for the password. Enter it, and it should give you internet access for a certain amount of time. Each time the time expires, you need to reenter the password.
If you don't already have the Internet Channel, then I don't think this would work since it needs an internet browser to enter the password.