My Solution

Heres a solution that may possibly help fix and repair Nokia X3 Failure of LED backlight  on keypad and LCD Lcd display. This solution completely shows the lines paths and particular components that may need to check and replace if the LED backlight failed to light up.  There is a fusible resistor is been connected to the LED backlight circuit in case a short circuit occurred if the device is being damaged by a liquid (Wet Damaged). See solution above marked with number 1.
That resistor is the one feeds the power supply from the phones battery.
If that resistor is open both keypad and LCD display LEDs will not light up for it is the main LED supplying positive voltage.
The negative current or voltage has two circuits, one for the LCD (marked with number 4) and the other one is for the keypads (marked with number 3).
You may check / replace each particular corresponding  connected components highlighted on the picture for further troubleshooting.