My iPhone 5 started having this problem. It's an old phone that was long past it's expire date so I upgraded to the 7. 5S is a 2013 phone so if it's just starting to act up now, you're doing OK.
I can still get down to 1% with my 5s. Question: how much does it cost to replace one? Do you guys usually switch it out yourself? Is the replacement battery good?
I bought my 5S in summer 2015 and it starting acting up like half a year ago (atleast, if not longer). It is a 2013 phone and I guess with new updates but it still didn't last me as long as I would have liked considering my 4 last me 5+ years.