Hmm, it sounds like the small round cable that you mention must be to hook up to some sort of external power device as the monitor is clearly designed with portable gaming in mind.
The best thing to do if you're planning on playing through a normal tv set is just to remove the monitor altogether, I think they simply click onto the top of the Gamecube from memory so it shouldn't be difficult to remove.
Hopefully once the monitor is unclipped it will simply be a matter of connecting it all up to your tv set.
Hope that's helpful, will try to advise further if you're still having problems.
Oh and welcome to the forums by the way