ok you need a a router. Not a wired or wireless. The reason why:
Because you need an ethernet cable going from your LAN adaptor, which i assume is plugged into the wii, to your router and then your router is to be plugged into the modem. Im sure speedtouch is the little tiny thing that is from BT? If it is then yeah i had that ages ago and basically you are screwed until you get yourself a wiered router at the least.
If you buy yourself and wireless one then you can search for an access point on your wireless router directly from your wii and if all goes well it will connect.
Hope that helps