Here's some hints on how to fix Nokia C1-01 light failure problem:
1) Check the connection between the inductor coil and the VBAT 3.7V line paths.
If this is open or cut, the LED light is lack of power supply to it.
2) Check the inductor coil for possible damaged, this holds the current stability of the LED light replace if
3) Check all the connection between the Keypad white LED light and the LCD light connection from in
between the component where it is being connected.
4. If all the connection were okay, check the transistor and diode. Replace if necessary.
5) If all above the above mention seem already been taken but seems the problem still exists. Check the LCD white light bulb LED. (there also a possibility that the LCDs backlight is damaged)