Any hope you had of playing Super Mario Sunshine or Super Smash Bros. Melee on your Wii U can be cast aside. Along with every other GameCube game on the market, they won't be playable.
Giant Bomb has learned from Nintendo PR director Mark Franklin that Wii U is not compatible with GameCube games, which explains the apparent lack of GameCube controller inputs on the system. On the bright side, we do know it's fully backwards compatible with Wii games and controllers (including things like the Nunchuk and Balance Board), so your existing collection of Nintendo products won't be entirely useless on Wii U. The Wii, by contrast, had four GameCube controller inputs and was able to play all of its predecessor's games.
The system is planned for a release in 2012, at some point after March.