The easiest suggestion would be to get a wireless (compatible) router for the Wii. If you're not willing to spend that money, then keep the LAN adapter, and hook it up to your wired broadband modem. You'll also need an ethernet cable. When that's hooked up (make sure everything is assembled correctly), go to the Wii setting/ options and go:
Wii Settings > Internet > Connection Settings > New Connection >Wired Router
Thw Wii will then initialize a connection test. Hope it works.