Same thing happened to me for a new Asus x205ta ordered on Amazon. Worked fine first day, but then would not power on evern after messing around with it for about 1/2 hour. After completing the return request on Amazon on my phone, I turned the X205TA on its side and pressed the power button wile pressing on the bottom of the case under the power button (like a pinch with my thumb and forefinger). For reasons unknown, it powered back up.
I am still sending it back because I think it is something defective witihn this particular unit. I was considering just getting a replacement, until I saw this on Google where others have had this problem.
Mark_32 :
I have the same problem
Except mine hasn't been opened and is only a week old
I shut it down before leaving on vacation and now it doesn't work anymore . No screen but my external hard drive does power up when connected
Can't send it back because I'm on the other side of the planet and I really need the laptop