We had the same problem during a period of 1 month, starting not starting then would not start anymore, tried to find answer on internet, but no good one, took out the plastic behind the keyboard, no screws, you only have to bend up the plastic, there are some links on internet showing how to do it.
https://www.youtube.com/watch?v=Y-35mUbeQnQ (where you can see in the end of the film how to bend out the plastic cover behind the keyboard. No screws has to be taken out)
Tested the push button and it was broken. Made a short cut over the push button with a small cable just to be sure to see if the computer started and it did. We changed the push button which can be soldered from top of the pc board without taking anything apart except for the plastic cover mention before. After we soldered a new push button the computer has been starting every time after. It is more likely the power button itself is broken than the power plug or power supply as many are suggesting on some site on internet. New push button you can buy in any store who sells electronic components and the cost is around 1 dollar. More info at
http://www.nasab.com/PDF_Service/Esprim ... mputer.pdf