To fix the dark screen display problem, you may first try to replace or install a new LCD screen module, to easily confirm if the LCD's backlight LEDs were still working or already damaged.
You can also check the LED's using the multimeter if you can manage without damaging the LCD flex connector.
If the newly installed LCD still shows the same problem and the LED light doesn't light up, refer to the solution below and locate the highlighted components on the PCB board.
Check the filter resistors and capacitors connected to positive and negative supply voltage of the LED pins.
Trace also each connection line paths were it is being connected to a certain components such as the resistors. Check and replace also the chip transistors.
If both keypad LED isn't working also, check the VBAT battery voltage on the filter coil and check also the coil for possible damaged. If the voltage reading is okay, replace the LED driver IC, for it also control the supply voltage both on LCD and Keypad Light LED.
Nokia 2690 Display Light Problem Solution