I think I have found a solution for this!
Now this might sound strange, but it worked for me, so it's worth trying:
1. Go to Settings - General - Background App Refresh -> And turn this option OFF
2. Let your iPhone drain itself until shutting down by itself - Important!
3. Recharge it again - it will flash an empty battery icon for a couple of minutes until booting up. This is normal.
That's it. problem solved for me!
My iPhone 5S was losing 1% of battery every couple of minutes. I ended up with no battery at 1PM !
So, I've recharged it, and so on. At the end of the day, I did not have a charger with me, so I just waited to get back in my car, and the iPhone just shut itself down.
The Background App Refresh setting was turned off by me as a "last resort" move to save just a little more battery... But now I keep it like this (Who needs this anyway?).
I hope it will work for you guys too.
(let me know out of curiosity )
Nir.