Iphone 4S is displaying black screen but is still on. Can't restart because lock button is broken
It sounds a lot like that to me. You could disconnect and reconnect the LCD cable yourself, if you are comfortable with opening up the iPhone. You would need a special iPhone screwdriver (one with 5-point star tip, AKA pentalobe), which can really only be found online.

You could also take it in to a repair shop, if any are around you, but whatever you choose don't over pay, as a working iPhone 4s is only worth around $100.