If your phone has had a broken power button for awhile and you let the battery drain often to the point of the phones turning off then there is a chance your battery may need replaced. Lithium ion batteries will have a shorter lifespan if a person let's the battery drain below a certain percentage often (below 20-30%), especially if it's to the point it shuts the phone off. I know you mentioned that the battery still had plenty of battery left but the software that monitors the battery percentage may be inaccurate at times especially if your battery has an issue. Since plugging the phone into the charger does not power it on eventually like it should then I would start by inspecting the charge port for damaged or missing pins or if it's dirty and needs cleaned. If the charge port and your charging cable appear to be ok then you may consider checking the battery next. You can use a volt meter on the batteries connector which plugs into the motherboard to check it's voltage if desired. Or replace the battery. You did not mention if you have tried plugging your phone into your computer to see if iTunes can recognize it. If you can get iTunes to recognize the phone then it could be a software related issue or possibly a screen issue.