I was going to ask questions in the comments, but I realized you were asking for information on how to diagnose it, as opposed to asking what the problem could be. As you have been told, a flashlight is a great diagnostic tool when attempting to see what is wrong with an LCD monitor.
You would also test all the available connections on the monitor. Don't limit yourself to only using the VGA connection. If the monitor has a DVI or an HDMI connection, test them as well.
Remove the bezel around any menu buttons, and remove the plastic button covers themselves. Make sure that none of the buttons are being pressed in by the bezel or the casing... as well as make sure none of the buttons are broken on the printed circuit board they are connected to.
Use different cables. After you have tested all the ports, get different cables and test them again. Test the cables on other monitors to make sure they work properly.
Also, as pointed out, if you find out that there is something being displayed, but just that there is no light to illuminate it, it would either be the CCFL illuminating the screen that isn't working (low chance), or it would be the inverter that supplies the high voltage necessary to kick-start the CCFL (high chance). Inverters are almost cheap enough to purchase on the off chance that swapping it out will cause a bulb to light again.
After that... you would check any cabling behind the LCD panel itself... but unless the unit was dropped, it's not likely the internal cabling would be bad. So, after this, you really start needing a multi-meter to check things.