Due to the age of the machine, along with the backlight coming on for a few seconds after turn on or wake up 90% chance the CCFL bulb in the LCD has reached it's lifespan. The other 10% is the inverter or on the logic board. The CCFL bulb can be replaced, but it is not an easy job. Most people chose to replace the entire LCD. New LCDs for these are available on eBay for about $70USD including shipping.
To test the inverter cable. Preform a continuity test using an analog multimeter/multitester (One with the needle.) while moving the wires around. Do this for each wire in the cable. If the needle sweeps hard at any point, it is a break in the wire and the cable needs replaced. Digital multimeters usually have to slow of a reaction time for this type of test and/or have circuitry in them to ignore quick fluctuations in a reading.
One way to test the inverter, if you have a multimeter that can measure frequencies of 50k or more, take a piece of paper and use it as an air gap between the probes and run it along the lower bezel if you get a measurement of around 30k towards the center your inverter is good. Inverters are cheap enough to buy one to swap out to be sure it is not the problem.
NOTE: The following is for reference only.
With the upper case off when you look at the backlight header/socket on the logic board there are 4 pins to it. The first one to the left is pin #1 and should read 12VDC or a little more (This is the supply line.). Pin #2 should be 5VDC and is what opens the gates on the inverter. Pin #3 is ground - Since the backlight is controlled by PWM (Pulse Wave Modulation) it is actually connected to a tank circuit and with power off may show a difference in potential difference from chassis ground. If the inductor or capacitor is bad in the tank circuit you will only have ground during half a wave cycle. If they are both bad you will have no ground, unless one of them is shorted. Pin #4 is the control voltage and should read 5VDC when the backlight is set to full brightness and something less than that on lower brightness levels. Unless you have special probes for your multimeter/multitester I would only test the voltages on the outer two pins (pin #1 and #4). Most test probes are too large for testing the inner pins (#2 and #3) and risks shorting out parts on the motherboard - and/or inverter, if hooked up.