If you have someone call your phone, does it still vibrate/ring? Or does your computer recognize it when you plug it in through USB? If so, then you'll probably just need to replace the screen assembly.
Find an LCD/Digitizer assembly for the 5S, iFixit sells good quality ones (the cheap ones can cause major problems and in my opinion are a waste of money). I doubt the power outage caused it, as it sounds like the LCD is damaged. Black lines and splotches are typical of a physically damaged LCD.