Scenario 1: You need an Apple OEM cord and wall plug. I didn't believe this theory the first time I heard it. But as an owner of a repair shop I invested in one to put the issue to rest. It does work and everytime I repair an Apple device i plug it into an OEM charger and it comes right up no problem. Why? OEM sends a signal to the devices informing it all is ok. It will not connect to computer in this state.
Scenario 2: Battery is disconnected or bad charging port/disconnected charging port. You may want to check all your connections again. If they seem fine try replacing the charging port. You would be surprised how easily they go bad. This would explain a disconnect to computer.
Scenario 3: Short in the Logic board. This is an...