I have the same EXACT problem you have on a Dell Inspiron 13-7378 after upgrading the BIOS to 1.26.0
The F10 key works and adjusts the backlight slider in the Windows Mobility Center but the backlight never comes on. It does not come on at startup either. They Caps Lock light does come on. I checked the backlight cable seating on the keyboard daughter board under the battery and it is fine. Did you find a fix to this? Can you post it?