Good question. For this particular model, the back light is integrated into the LCD panel. The issue lies at one of three points: 1) the LCD panel itself has gone bad, 2) the cable connecting the LCD to the system board has gone bad or become loose, or 3) the system board has stopped sending video to the LCD connection. Just because video is available through a VGA port doesn't mean that the system board is perfect. I would start with the simplest solution, which is to check the cable itself. Even if the connections are solid, you might want to unplug and re-plug them. I don't recall exactly where the LCD cable is within the base, but I believe it plugs in by the fan and can be adjusted without fully disassembling the unit. At the other end, there are two screws at the bottom of the LCD bezel which need to be removed. The bezel will then snap off (be careful not to break it) - revealing four screws that hold the LCD down against the cable. Good luck!
Unfortunately, there's not much you can do in this circumstance; it's a problem that most aging 6930's see, and HP has since made design changes to improve the latching mechanism. You could try replacing the base cover with a newer one, as that assembly should be included, or try bending the spring-pushed latching bar to have a better grip. Both require a considerable amount of work, tearing the machine down to it's core, and aren't guaranteed to resolve the problem. The new LCDs have thinner clips that latch easier, and the internal latch bar is metal instead of plastic. I, sadly, had to replace the base cover on about 2000 machines in 2009-2010. Good luck!