Backlit screen but no display.

I had the same problem with my Nintendo Switch. Try this remove the ribbon cable from the game card slot that is connected on the left hand side. Next grab some rubbing alcohol and clean the ribbon cable and the pins with the alcohol. (Careful not to use too much of the alcohol as you might get the LCD screen or other components burnt). Last reconnect the ribbon cable back into the slot and turn on the system. This is what I did to my switch not 100% sure if it will work with all Switch's.

One of the biggest fixes that I’ve been seeing around is simply disconnecting the Joy-Cons and plugging up your usb-c charger to give it a direct charge. Doing this allows the switch to be directly charged without any interruptions from the Joy-Cons. A lot of people report problems with charging with the Joy-Cons plugged up. There’s been a shockingly huge amount of people that has used this method to bring life back to their Switches. Also remember to use either your original usb-c charger or a certified aftermarket one. Make sure to give it enough time to charge as well. A seriously dead Switch can take many hours to even turn on. Good luck.