Usually these things switch based on a seeing sync pulses - in particular the vsync pulse (pin 14 if it's a 15 pin vga socket). So the first thing you can try is to try building something similar to the PWM I used but set to say 60 hz and see if that triggers the device to turn on.
TBH a good read about the VGA socket and the various modes would be useful, iirc they require various signals to figure out what's going on, you may need to also generate horizontal sync...
Presumably your board is all one unit, i.e. controller, psu and backlight on a single board?
Can you get a circuit diagram for it?