Depends on how savy you are with electronics. No point you trying to fix it if you can't get parts due to age of the laptop and can't get around it by making a connector or soldering. If that is the case, wait till it stops working again & get a shop to repair it if you need it, or you could risk a repair if you don't 'need' it.
If it was me and I owned it, I would take the bezel off around the LCD (if that is where the inverter is in on your laptop, was on my old toughbook) and carefully try moving the wires/ribbon coming in to the inverter. If the backlight goes out, you know where it is. Then I would patch it up. Oh, if you attempt this, careful of the wires coming out of the inverter. I believe they are fairly high voltage. Oh,and if it somehow ends up being the CFL, you can order generic replacement tubes, I think for around $25, but it is a task to put them in.