Hey Linda, here are some troubleshooting techniques iFixit recommends users who have this power issue to do:
Power Cord Problems
If your power cord is plugged in but the GameCube will not power up, unplug the power cord from the back of the GameCube. Check the port to see if in is dusty or dirty. If it is, gently blow on the area to remove the dust and replug in the power cord.
If the GameCube will not turn on, check on the left and right sides to see if the fan port is clogged. On the sides there is a gridded square that allows air to flow through the console to cool it. If the port is covered with dust you may just need to clean out the port to allow cool air to flow through the console again. You may need to repair or clean the fan. Also, to decrease the chances of this problem happening again, try to keep your GameCube in an open location where it will not easily suck in dust.
Optical Drive Failure
If none of the above options work then your Optical Drive may have malfunctioned. The optical drive is the drive that reads the disc and transmits the images to your TV. If your GameCube will not turn on, your optical drive may have malfunctioned. You will need to buy a new optical drive and replace it. To replace the drive follow the Replacing the Optical Drive Guide.
If none of the problems above seem to fix your GameCube, it may be a Motherboard malfunction. In Nintendo Products this problem is extremely rare so be sure that none of the solutions above work before you try this. You will need to buy a new Motherboard and replace your old one, following Replacing the GameCube Motherboard Guide.
-Let me know how it goes.