Now that you have it closed up, plug it in, and try to turn it on. It should function normally. If it doesn't turn on, you may have a bad connection...
Picture this; You wake up in the morning, and you try to turn on your gamecube; you push the button, and you hear a whirring noise, but nothing else happens. NOOOOOO! the guys are coming over to play Super Smash Bros. in a couple of hours, and the gamecube doesn't work! Before you launch into a bunch of "that's what you get for buying a gamecube. loser!'", I know, I don't want to hear it. For those of you who love your cube, though, I hope this helps.
Step 1: What went wrong?
If there has recently been a power surge where you live, that may be responsible for the problem you are having. The symptoms are as follows: the external power supply functions normally, but when you plug in the cube and push the power button, the fan turns on, but nothing else happens; no light, no video, no spinning disc, nothing. Nothing you do changes this condition. So what's wrong? is the gamecube fried? The Nintendo maintenance procedure for fixing this problem is to replace the entire board, which, obviously, is expensive and wasteful; the only part that's fried, as it turns out, is the fuse, which, for reasons I will never understand, Nintendo went out of its way to hide deep inside the system. I'm going to tell you how to take the system apart and replace the fuse with a cheap, equivalent fuse from radio shack or another electronic parts supplier.