For your 5S, I would hold the home button and sleep wake at the same time.
Tried that, goes back to the same screen as in picture above.
Try recovery mode, make sure your computer is handy:
Use recovery mode
Turn off your device. ...
Plug the device's USB cable into your computer only.
Hold down the device's Home button as you connect the USB cable to it.
When you see the Connect to iTunes screen, release the Home button. ...
Use iTunes to restore your device.