To solved this problem, you may first try to replace and install a new LCD module to easily determine if the LCD LED's were still working or not. If the newly installed show same problem, refer to the following solution below and check some components that holds for the LCD backlight LED's.
Nokia 1680c,1681 Display Light Jumper Solution
Check and clean the LCD pin connector, for possible damaged and oxidation build up. Trace each corresponding LED connection down to a components where it is connected, check also and then replace that components if found damaged. Check and read the LED driver supply voltage and the filter coil. If these voltage is missing the keypad LED will also not work. If both LCD and the Keypad LED doesn't work, replace the LED driver IC.