LCD backlight is merely a cold cathode florescent tube (CCFT), or cold florescent light (CFL) tube - it either works or it doesn't. I'm inclined to think that your inverter is failing (it hasn't gone completely since it worked for a second...)
How many backlight tubes are in your LCD (count the number of wires that originate from the LCD to the inverter and divide by two)? If you have more than one tube, then the chances of both tubes failing (without actually being broken) is really nil nil. If you have only one tube, then the chances of your tube failing is still pretty darn close to being nil...
Of course, you can also meter the two contacts on the inverter - where the backlight plugs in - you oughta get somewhere around 1500V. Used to know someone who tested inverters by sticking his finger at the backlight connector - if you smell burnt flesh, there's nothing wrong with it.
Unsolved 
