Thanks for all the responses everyone!
I still haven't been able to fix it. I went to Home Depot and got some spade connectors (I think that's what it's called). They fit very snugly on the blades where the PTO clutch is. I connected the wires, but still no dice. I switched the wires just in case I had them backwards. No luck. While I don't have any way to test this, I'm pretty sure the connection that wasn't being made when that wire was disconnected is now connected.
I highly doubt that the wire that was hanging loose was hanging loose before and everything still worked.
I highly doubt that the wire came loose AND something else broke at the same time (unless there's some fuse or something).
I am at a lost. I think I'll be able to stop by a John Deere dealership next week, but any other suggestions in the mean time?