You missed the easiest and most common solution to this, most of what has been said above is not correct. when you change the screen odds are you will have removed the little magnets, if your power button only works when long pressed then this is why, simple fix, go into settings and the lock/unlock and turn if off. FIXED
Have you treated the logic board for water damage? if not that might be your problem. Where to find one is simple...eBay. but as said previously the logic board is the phone really..
I was going to suggest the volume + power button to bring up the hidden menu if android but as you say it doesn't have one.. I used to deal in tablets, china tablets, I would suggest contacting the manufacturer and asking them for the firmware so you can restore the tablet. Back when I was selling them the factories I got them from sent me the firmware for the tablets and one of their chinese programs I had to use to restore the tabs that had problems, might be worth contacting them :)
