If you close and open the laptop without powering down (making sure it's not set to go to sleep or standby, and the monitor is set to turn off when you close it), does the light come back on briefly before switching off again? If so, it sounds like the CCT (what actually makes the light in a many LCD screens) has gone out. Unfortunately, while it can be replaced yourself, I would advise against it. If you accidentally snap the tube, you don't want to breathe what's inside. If you don't feel like buying a new monitor for it (either replacing the whole bit yourself [safer than just the CCT, but more costly] or sending it in for repair [even more costly]) and don't move it around much, most laptops have a VGA port for plugging in a desktop monitor.