My iPhone has a black screen. What should I do?

Uhm hello.

Sooo maybe like 3 weeks ago I was charging my Iphone 5 and I put it into airplane mode and low battery mode when I was about to take the charger off I accidentally dropped it and when I checked it, it has a black screen. People say that if you hold the power button and home button it will solve your problem but, sadly, my power button is broken. When I plug it into my laptop it doesn't recognize my phone so now I am waiting for it to drain (up until now) and it is taking sooo long. How long do you think it will drain if it is set into low battery and airplane mode? I don't know what to do. Please help me :(

(My phone's screen has been replaced before I dropped it)