If you already tried replacing the LCD module but still the problem still exists, you may refer to this solution below for it shows some tips on how to fix the problem.
Locate the components below on the 2700c PCB board and do some test and check up procedures.
Note: This solution may works if the keypad backlight LED is working and only the LCD has no illuminating light on it.
Nokia 2700c Display Backlight LED Problem Solution
From the LCD pin connector trace the display backlight LED positive (LED+) and LED negative (LED-) supply voltage connection line paths down to the components where it is being connected. There is a test spots on the PCB where you can also trace its connection.
Then check also the resistor and the capacitor near the LCD pin connector and then replace if found faulty or damaged. You don't need to check the LED driver chip unless if the keypad LED is working OK.
If both the keypad and the display backlight doesn't work, the problem may possibly lies on the LED driver IC. Check the 3.7V first on the filter coil before attempting to replace the LED driver chip of the 2700c.
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