To fix LCD backlight problem on the phone, you may first try to install a new LCD screen module just to easily determine and confirm if the backlight LED are still working or not. Since the LED's were mounted at the back of it.
If the new installed LCD still shows a dark screen and the back light won't turn ON or glow. refer to the solution below and do have a check and tracing procedure on the display light components and its connection line paths.
Nokia 1508 Not Working Display Backlight LED Solution
The LCD display light has a LED driver used in it, you may first check the LED positive and negative supply voltages connection line paths from the output terminals of the chip and corresponding components down to the LCD pin connector.
If all the line paths is okay, check the battery supply voltage (VBAT) on the LED driver. If the voltage reading is okay, there is a possibility that the LED driver is faulty or damaged. replace it if necessary.