iPhone won't turn on
Unfortunately you belong to a very large, very angry community.

That community > 800,000 has decided to sue Apple over this issue.

You can wait for the outcome of the lawsuit, or have it done for $150, or try some DIY for $5 for the part.

One other way is to try to clean the stuck power button.

Use a can of compressed air (for computers) found in computer stores and blast it. You can also try to flood it with some alcohol (ethanol or isopropanol) to try to wash out debris.

To be more effective, removing the back cover (two screws) could help.

If neither of these worked then you can consider replacing the power button.

There are plenty of online instructions on how to replace the power button.
I won't fool you, that procedure is not easy. It is time consuming and requires lots of patience and skills + tools... that is why the angry mob. :)