Here's the Nokia 1508 solution for not working keypad back light LED problem. This solution shows the connection line paths of the keypad light that may be useful for troubleshooting the problem.

To fix a not working keypad light, you may first check the two LED's on the keypad matrix. These two LED's were connected in series connection, it means if one LED is bad the other one will also won't light up. So ensure that both LED's are working.If the two LED lights seems both working okay, refer to the solution below and do have a voltage reading and connection line paths tracing procedures.

Nokia 1508 Keypad Light Jumper Ways Solution

You may first check the battery positive supply voltage (VBAT) which is about 3.7V connection, then the negative line paths which has a resistor connected to it. Check and or replace the resistor also if necessary.The LED controller IC is the one that holds and engaged to switch ON and OFF the keypad led, it work as a switching mechanism of the keypad light. There is a possibility that the problem may also lies on the LED control IC and you may need to rework or replace this said chip just to fix such 1508 keypad light problem.