Why did you pay $80 for GTA V? MSRP is $59.99 and I got mine for less with old trade-ins.
Anyways. I am assuming you purchased a used one. When investing in used items - you should expect things to fail. If you wanted new quality you have to pay for new quality with the manufacturers warranty.
The component that tells Xbox the tray is closed is broken and needs to be serviced by someone who knows what they are doing and aren't jamming things around.
Sorry to hear the tray is broken. Perhaps ask the previous owner(s) or bring it back to the store you got it from and see if they can help.