I had the same problem with a 27-inch iMac 2009 i7, I changed the inverter board several times without success. Then removed the flat cable between inverter board and display, which I had previously renewed and curiously, the backlight is now permanently on.
Although the temperature sensor cable of the LCD was also defective, but had the consequence that the backlight did not even concern. The temperature sensor is a standard component, a 2N3904 transistor and easy to replace.