backlights, either CCFL or LED. Your LCD is fine it sounds like, I believe these had LED backlights and a backlight fuse, its most likely the connector but it could be the fuse. https://www.youtube.com/watch?v=u1ZGhd2A...
I would have thought the battery was faulty. Tablets usually can’t run without a working battery even when plugged in. If you don’t want to spend money on a new battery to test that idea out, you might be able to connect the proper voltage from a DC bench power supply to the B+ and B- solder pins to see if that works.
Does the disc drive start moving with a disc in it? If it stays powered with the fan on I doubt its anything to do with the power supply. It could be a loose LCD flex cable, laptops tend to have LCD connection problems after years of use. Try connecting it to another monitor with a VGA cable if it has a VGA output to project to another screen. If that works then its the LCD or the flex connection. If not then it could be a fatal motherboard issue.