iphone 5s would turn off and on repeatedly then work fine once i plugged it in. now it shows the dead battery icon, flashes apple logo, then a black lit screen

two weeks ago i dropped my iphone 5s and it had a single crack from it. the fall was less than 2ft and the phone was acting fine. then a week later it began doing a mixture of things:

• Shutting off whenever i would pull up my camera or begin video taping (including snapchat)
• shutting on and off repeatedly then showing the dead battery sign but once i plugged it in it turned on
• once unplugged from charger, sometimes would immediately turn off again.
• even after a full charge it would say it would shut off with 92% batter
• Now, it wont turn on unless plugged in and when it is plugged into to the charger, it shows the dead battery sign for a brief moment. flashed the apple logo for 1 second then is 'on' but screen is black.

I've attempted a hard reset holding both the home and lock button, DFU mode, recovery mode, ive attempted restoring it on itunes and updating the software but it wont allow me because i cannot turn the phone on. Any help would be awesome. --Thanks