Okay, I can officially say this problem is sovled, and I needed not a trip to the Apple Store nor a replacement part. The problem was a connector on the LCD. If you look at the LCD back, on the bottom right there is a fpc connector. When I looked at it, it was half disconnected. I plugged it back in, put the LCD back on and sure enough it works now as I am typing on it. If you have a iPad Air LCD with no backlight, check that first.