That "box" you have is a DSL Modem. Not a router. You need a wireless router. I recommend this one:
It's not hard to set up. Here is what this will do for you:
-Allows your Wii to connect to the internet wirelessly
-Allows multiple computers to connect to the internet in your house
-Allows for networks within your house
It connects to your router by ethernet cable. If you get it and set it up it will look like this: Phone jack -> Modem -> Router -> Computers/Wii
EDIT: To answer your main question, the WiFi USB adapter would give you internet, but it is a waste of money and would probably be a lot slower than a wireless router (which can do 20 times what the adapter can).