Phone crashed -> battery low + display problem. What could be broken?

4S got run over by a car.

I changed the broken display, functionality (sound, vibration, cameras, wireless, mobile data, emails and other notifications come in, I can here SIRI, etc.) is working (tested via cylay).

Battery always low and phone turns off pretty fast after unplugging from the charger. Plugged in it turns off itself after about five minutes and restarts immediately again + display doesn't work (sometimes gets bright but there appears no clear picture only some grey tones). It worked about three times though that the display was clear and I could use the phone for about five minutes before it lost its picture again. What could be broken?