Here's some hints on how to troubleshoot the "Nokia X2-01 LCD display backlight not working problem".
" Note: ensure that the LCd is okay or try replacing it before proceeding"
1. Check the LCD pin connector for possible damaged, resolder if necessary.
2. Trace each of the LED + (positive and LED - (negative connection line paths down to the resistors where it is being connected.
3. Check the connected resistor, replace if found damaged.
4. Check the 3.7 volts battery voltage (VBAT) across the inductor coil, check also the coil for possible open circuit.
5. If the voltage reading is okay and the coil is good. Replace the the LED driver IC.
Nokia X2-01 LCD Light LED Not Working Problem - Solution