Hello I will help you with your question,
What happens when you turn the spare tire lift with the tire tools?
It sounds like the lift ratchet has rusted so it will no longer catch. You can try spraying it with rust spray and letting it soak (really flood it) and see if it comes back to life. Otherwise you will have to replace the lift. If it is raised too high without the tire in place it can get stuck all the way up, I realize this is not your problem but I thought I would warn you of that.
The service information to install the tire and replace the lift is below, maybe it will help but I think you will have to replace the lift due to rust.
Click here for procedure
Look this over and let me know how i can help