This solution worked for me. Thanks to author of the post:
I have this same problem since replacing my charger. It does not do it with the charger built for the regular surface with less amperage, but it does it with every after market standard charger I buy. The solution: Hold down the upper volume button, then also hold the power button for about 15 seconds, then let go of the upper volume button, and then let go of the poser button. Wait a few seconds, then it should start. If not, press the power button. After it starts, plug in the power. Then go to Amazon and buy the smaller charger for the regular surface model (Assuming you have the Surface Pro 2) they have the same plug. It may take a little longer to charge (Although I have not noticed it) but it does work fine on mine.