To fix the not working keypad Light, you may first check the two LED's using the multimeter. The LED's is connected via series connection so therefore if there is one LED is busted the other one will not light up too. So replace it if found damaged.
If the two keypad light seems both working, you may refer to the solution below and do some test and check up procedure on some components highlighted on the image.
Nokia 2690 keypad Light Problem Jumper Ways (Tracks)
The LED Driver chip also holds for the LCD display backlight. if the LCD's light is also been affected, you may consider that the problem lies on the LED driver chip.
You may first check the filter coil and the voltage supply (3.8 battery voltage) across to it., before trying to replace it.