Hi, I’m Louie. Sorry you had to wait awhile but I’ll do my best to help.Initially, bear with us; re-confirm and post back with the complete model number of the phone.
Appreciate the additional information.
Proposed preliminary tries (if have not been done):
Please post back results / observations of the above once tried.
Would you be comfortable forcing the iPhone 5S into Recovery Mode ?
If yes, please be aware: Recovery mode would DELETE all the data on the phone. Consider if and only if a recent back up on iCloud or a computer has been made.
May have to be done a number of times.