On a scale of 1-10, how addictive is Wii Fit? I was looking at it the other day, but to spend the money on a Wii and the Fit package, I would need to be assured that I'd keep using it and that would require a factor of addictiveness.
If you remove the one in there now and hook your 360 up to it, it will disconnect whatever is plugged into it now from the internet. I assume it's a desktop computer.
What you can get is an inexpensive network router. A salesman at Best Buy or other consumer electronics store will be able to find what you need if you describe your problem.
With a router, you can connect multiple devices up to the same internet connection and share it.