Be careful in there - if the inverter happens to be working, it will feed about 1 kV to the actual light tubes! Otherwise it should be pretty obvious where it sits if you get the monitor apart. It usually just takes 12V in and there can be 1, 2 or 4 tubes that it is supposed to light up.
If you can see the display operating with a flashlight, I'm pretty
sure the problem is either the inverter, the CCFL tube (if there is only one)
or the power control of the inverter. This is a problem common enough to motivate webshops like this!