First check inside the headphone connector with a strong light and magnifying glass. Inside there is a moisture indicator. It is a piece of paper and if moisture got inside the paper will turn pink on one half of it. Moisture will cause the problem you are indicating. Second if no pink stripe is found try restoring your iPod in iTunes. Make sure you restore in not just sync it. Starting fresh with erasing, restoring and updating (if there is a OS update) will fix some software glitches. Lastly take a dry old toothbrush and make sure there is no debris in the dock connector area. Debris in this area can short two tabs on the connector together and cause problems.
You didn't state how you tried to power on so I will assume you did just by pressing the iPod power button. I just went through replacing the front of my iPod touch. I had to do the whole process twice because I am having an issue with it. Long story, short, mine would not power on either time without plugging in the usb cord that was connected to the wall outlet. Try that.