Well first, are you sure it is charged? Fully? If you believe it is, I would suggest trying to remove the battery, for about 20 seconds, and placing it back in. Then trying to restart the phone again. If it doesn't start, the battery may well not be working/or not charging. Even though it seems to be. You may want to try a different cord for charging and also a different battery.
You should also check to make sure the power button isn't stuck. Or any of the other buttons for that matter. Any stuck buttons can keep the phone from turning on.
If all else fails (no stuck buttons, battery is fully charged and functioning correctly as is charger) then you may want to try a "Hard Reset". However, I say that only as a last resort option, as doing so would wipe your data.
If even a "Hard Reset" doesn't fix this, you last options are to either have a tech look at it and see if it is salvageable, or get a new phone.