Here's my suggestion. Try using a Game Cube disc once the unit is back together. Game Cube discs are a smaller diameter. Pick up a cheap one at a local game store just for testing.
If the Game Cube disc works, but the Wii still will not play regular Wii discs, I have a repair that works about 3 times in 5.
There are two metal flaps which move up in front of the inner disc slot to keep the disc from being pulled out of the drive.
Each flap should have a white plastic guide about 5 by 4 mm clipped to it. Sometimes this guide is broken, loose or missing.
I use a bit of heat shrink tubing about 5mm long to replace the missing guide.
After installing the new heat shrink guide, carefully bend the flaps with a needle nose pliers until they are close to (1mm away), but do not touch the edge of the Wii game disc.
Let me know if this works for you. Best of luck and hope this fixes it for you.