The exhaust vent coming off should not affect the dryer itself. I think that is a red herring. The dryer really doesn't care if it vents into the room or not. It has no way to tell. Vents detaching can sometimes indicate a blocked duct.

Start with the simplest/easiest.
Wiggle the plug and move the power cord around while someone tries to start it and see if you have a poor connection in the outlet or a break in the cord. Verify power supply is healthy at the terminal block on the back as well.
After that you are checking the components in the motor curcuit. So....wall->terminal block->timer->door switch-> thermal fuse->motor. I'd start at the terminal block then skip to thermal fuse as it is easy to access and can be more safely tested live if needed.

If it is not starting you can ignore the heating side of the system, it is not involved in whether the motor runs.