The connectors and flexes that connect the screen assembly to the logic board are very delicate. Follow the guide below and tell us what your phone looks like in comparison to step 15. (pictures are even better).
iPhone 5 Display Assembly Replacement
Losing a screw inside the device is also not good, especially if you closed the phone and powered it up. That loose screw could have caused damage as well.
However, based on your description, it sounds like one of the connectors was damaged. If it is on the flex side (screen side), then a new screen should solve the problem. If the damage is on the logic board, the it will require a micro-soldering repair.