Thank you for your detailed description of the issue you are facing with your iPhone 5. It appears that the main issue is that the touchscreen is not responding, despite attempts to press it or lift it up from the case.
It could be possible that the fall has damaged some internal components and wires, causing a disconnection between the touchscreen and other parts of the phone. It might also be possible that something in the case itself is preventing a connection between them.
The best course of action would be to open up your device and inspect all components for signs of damage or loose connections. If any are found, then they should be replaced or reconnected accordingly. You can also try using a different screen assembly or digitizer to see if those components are causing an issue.
If none of these solutions work, then it may require professional repair services as more serious problems may have occurred inside your device due to its fall.