My (unbroken) iPhone won't turn on!
You should get someone to open up your iPhone, disconnect the battery, and reconnect it. That's where I would start with it, it may be something more serious, but there's a pretty good chance just that the disconnection and reconnection of the battery on the logic board could work. You could try to do it yourself, but it requires taking off the touchscreen (digitizer) assembly, about a dozen small screws, and a few other parts like EMI shields. It's not too difficult, but if you decide to do it, you'll need to buy tools online and repair guides at iFixit.