Try updating it to a new firmware. I once had a 5C die in the middle of an update, and basically got it bricked, and had to finish updating it via iTunes to get it working again. Plug in your 5S to your computer and open up iTunes, then turn off your 5S, hold both the power and home buttons for around 10 seconds, release the power button but keep holding the home button until iTunes says something along the lines of "iTunes has detected an iPhone in recovery mode", and then you'll have an option to update it to the newest iOS, 10.3.3 or the iOS 11 beta 4 if you'd prefer that instead.