Can't turn on after battery replacement

The fact that you have backlight and vibration tells me the phone is likely still alive in there and is probably booting just fine, it’s just having a hard time communicating with the display. First thing to check is the display connectors, disconnect the battery and re-seat each one. If you still have the same symptoms, it is possible the ribbon cable leading into the display was damaged during the repair. At that point my next step would be to get another screen and test it on the phone. If the phone is still not working with another screen, there is a chance that the socket for the display on the logic board may have been damaged during the repair and at that point, you may want to start looking for board repair specialists in your area.