Default LCD screen keeps disconnecting.

I dropped my phone the other night onto a wooden floor, and the entire phone was still intact. But now the screen will sometimes just not turn on, but the bottom row of lights still turns on and I can unlock the phone and reset it. Sometimes the screen will work after resetting but then later glitch up. I have removed it from the case but couldn't get the LCD cable without feeling like it would break the phone. But there seems to be nothing wrong with the connection pins that I can view from the side.

Everywhere that I've rang to get repairs for it straight up says that it is a screen replacement, but I know that the LCD and digitizer still works and that the phone does turn on.

What's the best way to repair it?