Ok lets not get to screaming. Totally understand the moving the router point, perhaps you can leave an ethernet cable coiled up but accessible by an adult so its easier in the future.
You don't need to put the router in pairing mode. Only if your going to brigde to an ethernet extender, or you want an xbox to pair controllers etc, which you are not.
the hex key is an encrytion method. Some routers pad a small key, like "hello" to make it 64bit, of course you dont see this so its a pain and other systems just accept it. I dont know about the wii because I know my key is 128bits long so it just works.
Changing the key and perhaps switching to wep is the first port of call.
this link www.wiichat.com/nintendo-wii-online/5820-official-wii-wireless-internet-connection-guide.html shows you the commands on the wii.
if its any help tonight we are switching to going away and taking our wii with us, so tonight I need to pair it to another wireless router, happy to talk type in exactly what I did when I get there if thats any help, I know that router is using wpa.