Any significant events before the issue started? For example, liquid getting into the keyboard will usually fry at least one led; the keyboard backlight circuit on your MacBook model has 4 (or 6, I can’t remember) led’s wired serially, and fed by a ribbon cable supplying ~5vDC, modulated by a single chip that handles most things low current DC.
If the lights are fried, you could probably pick up a keyboard backlight replacement off Amazon or the like for maybe twenty bucks. It’s pretty much a drop in replacement, though you will have to gut the case fairly thoroughly to get at it.
If it’s the controller, the only real practical option is to replace the logic board, which really isn’t that practical, economically speaking. If you have a nice smt repair station setup and can find the chip (which you probably won’t be able to do, probably Apple proprietary, so perhaps a donor logic board) then you could replace the chip, but this is easier said than done, to put it mildly.
Dan’s solution will give you a fairly solid diagnosis. if you get a DC voltage of around 5V measured at the ribbon socket with the keyboard backlight fully on, then it’s the LED backlight sheet that’s bad. (You might need to measure AC and read the RMS voltage depending on your multimeter, due to the PWM driven circuit. I assume it would come out to something like 2V if you do it this way.)
if you get nothing... then it’s the logic board. Which, if it were the case, I would just learn to live without the backlight.
I’ll keep my fingers crossed that it’s the former.