I am an Apple unauthorised service provider that happened to get one 5s phone with the exact same issues as described here today, and here is how i fixed it.
1: Remove the entire display
2: Reconnect ONLY the ribbon cable with the HOME button
3: Connect the phone to iTunes "used a mac"
4: Push the POWER + HOME for 10 sec or as long is needed to get iTunes to see the phone and wanting to restore it
5: Do a full restore and wait until its all finished.
6: Once done, turn the phone off, and reconnect the screen and all its cables.
7: Turn on the phone, and you should have a working phone again.
:)