Here's what's going on:
1. The cable that connects to the engine is basically a kill switch. The other end is connected to the operator presence bar (the bar you release to stop the mower.) In your case, the cable has come loose from it's mount on the handle, so when you hold the bar down it doesn't put tension on the cable. See the white zip tie on the handle? It's hanging next to a plastic bracket. The plastic bracket has a large hole in the top, parallel to the cable. I'm not sure how the cable got out of that hole, but it needs to be going through that hole in order for the mechanism to work properly. Disconnect it at the engine (easier than doing it at the handle) and thread it through the hole. Then hook it back up to the engine. If cpurvis sees this, hopefully he can circle the parts I'm talking about. I'm on my tablet again and can't edit pictures.
2. The broken spring (yes, it is broken) has nothing to do with the engine not starting. It's part of the self propel mechanism. Unless the self propel systems starts acting weird (the mower doesn't want to stop moving) I would leave it alone. I think I disconnected that spring on my old Toro to make it easier to operate, and I don't remember having any issues as a result.