2007 Aveo LT Manual Trans won't go into gear

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So I have a friend's car that I'm trying to fix. She called me the other day and said that when she went to start it, it wouldn't start, and it wouldn't go into gear.

She towed it to me, and I just took a look at it.

A few observations:

• The clutch pedal doesn't go to the floor when pressed. It goes about 70% of the way, and if I really stand on it, I can get it to go another 20% or so. I can see marks in the floor that the pedal used to touch.
• Under the hood, there is what I assume is motor oil everywhere, but it's possible that it's gear oil. If I had to guess, it's motor oil leaking out of the valve cover gasket.
• There was some kind of sensor on the front/top of the trans that looked like it was broken, and it was hanging out. I think it was the one with the green and brown wires going to it. I tried to plug it back in, but I'm not sure if it's actually working or not.

If I really stand on the clutch pedal, I can push it down far enough to trip the clutch switch and allow the car to start. But no matter what I do, it won't go into gear while running.

See photo: