I've got a broken power (lock) button and Apple Store said there's nothing they can do. If your iPhone has turned off (either low battery, damage or an automatic shut down to refresh) and your lock button is broken, there's not much that can be done. There's just a slight possibility that it might turn on by itself after hitting 5% or more. Connect it to a power source (Computer, Wall Plug, etc.) If this doesn't work, your iPhone will have to be repaired (the lock button itself) but this is an expensive move. If it's severe, there's not much that anyone can do. If you turned your iPhone off by AssistiveTouch, I suggest you connect your iPhone to your Computer and go onto Apple Store, restore your General Settings from your iPhone (this can be done) and enable AssistiveTouch on your Computer. This could turn your iPhone back on. If this doesn't work, then you won't be able to use your iPhone.