Device is connected to wifi but no internet.

A common issue is pinching the antenna cables when you remount the logic board. While you were able to get a basic connection it is not strong enough for a full data stream to get across. Pop off the logic board and check your cable routing. You may need to replace the antenna unit. Also make sure you got the correct antenna onto the correct port on the AirPort board.

For your power issue lets get a better view of your batteries health and the charging logic. Install this gem of an app CoconutBattery take a snapshot of the main window and post it here for us to see Adding images to an existing question