Broken LCD Cannot Detect Touch Sensor

The LCD panel isn't responsible for touch input. There is an entirely different layer called the digitizer whose job is to recognize touch input. I suspect it got damaged along with the LCD panel but wasn't replaced.

 

See if the P button works in landscape mode. If it's not clickable in either orientation, it could be software which could be fixed possibly with a hard reset (hold lock button and home button together for 9 seconds). If not, it's a bad Dig. The good news is that you can only replace the digitizer/LCD as an assembly because they are fused together at the factory. You can find replacements on eBay if you were so inclined to attempt a repair yourself. If that's too daunting, let me know and I can provide you with more options.