Chosen Solution
If it’s driving the external monitor the logic board should be fine. I’d reseat the LDVS cable: MacBook Unibody Model A1342 Display Data/Camera Cable Replacement
It might be that something went wrong somewhere during the repair and you got something along the backlight circuit shorted. You should start with testing the backlight fuse for continuity. It’s the part number F9800 on your board..with a little googling you should be able to find the necessary references to locate it. It’s on the bottom side of the board, quite near the magsafe connector. Sorry but I don’t have any board like yours at hand to take a pic.
