Been there, done that.
DISCLAIMER: PLEASE BE CAREFUL. THE OUTPUT VOLTAGE CAN REACH PEAKS OF 1500 V.
As you said, probable causes are the CCFL or an inverter issue; however I would also have a quick look on the connections and the inverter supply voltage. Just to be sure, I would also try to validate if the On/Off and the brightness signals are at sane values.
Otherwise, I would test the inverter with a spare lamp, if available. If not, I would use a standard CFL bulb, removing his supply and connecting the CFL lamp wires to the laptop's inverter.
Most likely, the laptop CCFL is burnt out. Measure the length and diameter and order a compatible one on eBay. I was always able to find something close to me, but I'm not sure on the situation in AU.
PS: Keep the old CCFL's connectors to refit them to the new one.
However, please note that this is the way I would proceed; this message is not a recommendation or a how to. Take care when experimenting.