On the Macbook A1342 (LED driven display) the inverter signal (backlight lightning) goes thru the same cable than the video cable so there's only one socket and connector on the logic board for the two functions (video and lightning). The liquid spill could have affected this logic board area (upper right of the main board). I would remove the MB bottom panel (8 philips screws), disconnect the internal battery, remove the connector from the socket and carefully inspect it and report what you see. Also look for corrosion or liquid stains on the logic board.
Be careful with the display cable cause it's very fragile and easy to break if you don't remove it the right way. Check this iFixit manual and pay attention to steps 9 and 10.
MacBook Unibody Model A1342 Display Data/Camera Cable Replacement