iPhone not turning on

I'm not sure if I follow the possibility of a charging port being the issue. It sounds like the main potential problems are:

1. The Screen
2. Damage to the logic board during the battery replacement

The fact that the computer recognizes the phone indicates that the charging port works fine, and the fact that you can hear the sound coming from the device indicates that it is booting up.

I would try a replacement screen first. The LCD could have gone out or been damaged in some way. If that doesn't work I would assume there was damage done to the logic board during the battery swap, such as a blown backlight diode or blown backlight fuse. These repairs require microsoldering, but the components are pretty cheap.