How to jailbreak on an iOS 10.2 on an iPhone 5s
As per my little knowledge regarding iOS, I found Yalu jailbreak is updated and release to jailbreak iPhone 5S.


  Step 1: First up, connect your compatible iPhone or iPad device to a
  Windows or Mac computer using a Lightning/USB cable.
  
  Step 2: As a precaution, make a complete backup of the data on the
  device using iTunes so that you can return to it in case anything goes
  wrong with the jailbreak process you are about to perform.
  
  Step 3: Launch Cydia Impactor on your computer.
  
  Step 4: Drag and drop the yalu102 IPA file you downloaded earlier on
  Cydia Impactor.
  
  Step 6: Wait for Cydia Impactor to install the yalu102 jailbreak app
  onto your iPhone or iPad.
  
  Step 7: Once installed, disconnect your device from computer and head
  to Settings >  General > Profile(s) &amp; Device Management or General >
  Device Management and hit Trust and then Trust again on the
  certificate you used to install this jailbreak app.


For detailed tutorial, please check Redmond Pie or Team Cydia.