Maybe it's stuck in sleep mode, have you tried removing the battery and putting it back?
You can try the magnet trick, though don't know if it's the same like other 3ds models when the location is between the A button and start button just hover the magnet around that area and see if the screen shows an image..
Or it could be the backlight, to check go to a dark room and turn it on, if you see an image and it's just so dim, then it's probably the backlight is faulty.