As in title, I turn on my 2DS, the blue indicator light turns on, then the backlight for the screens come on, but the...

I have a similar issue, where I turn on my 2DS and the screens turn on, but stay black. But your issue sounds like hardware failure, maybe the motherboard was damaged in the possible fall. You can get replacement motherboards for around $20 or more, and there are a great deal of tutorials online on how to replace them. Good luck to you!