Touchscreen unresponsive unless pressed hard
I am trying to fix an iPhone 5.
The only apparent problem is that the touchscreen is almost completely unresponsive, to the point where I cannot unlock it.
I have trouble understanding the problem, for the following reasons.
The screen is popping out of the case from the two top corners. When I press very hard on all sides of the screen, and keep it pressed, most of the time I am able to get the touchscreen to be almost entirely responsive. Of course it has to be somebody else, it isn't doable with just 2 hands.
It thus seems that the screen and digitizer themselves actually work.
But conversely, if I open the phone, keep everything plugged but with the screen lifted up, and turn it on, it is still completely unresponsive. I saw on videos that theoretically a touchscreen would work fine if only lifted up and still plugged.
The previous owner told me the problem started after a fall, an indeed one of the corners bears heavy marks of that fall, and the case seems slightly bent.
However, if the problem was the bent case, wouldn't the touchscreen work when lifted up from the case?
Thanks a lot in advance for the help!