How to get iPhone to turn back on?

Replace battery first and check that you can find the clue what you have to do.

As Akash suggested, if you can remove the battery it will help. I would lift the screen and disconnect the battery, then plug the phone into a computer with iTunes installed. If the phone boots up with the battery disconnected, then the battery is likely the problem: follow this guide to replace it. If it doesn’t boot up, you should hopefully still get an error message of some description from iTunes. The error message should give you a better indication of what is wrong.