With pin 1 feeding +12V and pin 5 to (-)ground, probe with your voltmeter the exposed pins of the components nearby the black connector, that is the black rectangles and the round silver capacitor for +12V presence, come back with results.
Do you mean the display has never had backlight before or only since failed condition ?
The lamps can have +12V taken from the board and by grounding pin 6 can turn on; or reverse, the lamps can have (-)gnd from the board and by feeding +12V to pin 6 can turn on. A transistor on the board could also switch them on. Unclear how it works.