Update 7:
Background:
My daughters iPhone 5 started acting erratically last month after the 10.1.1 Update. Before the update it would maintain power in standby for somewhere over 24 hours. After the update to 10.1.1 it looked like it was draining rapidly going from 100% charge indication to somewhere in the 30's % range. It would then get the spinny bars and go dead.
I did a Recovery Mode Factory Restore with 10.1.1, but the problem remained the same.
We decided to update to 10.2.x to see if that might fix the issue, but the problem became more exaggerated with the phone indicator almost jumping in chunks down from 100% to the 40-50-60% range in a varying manner and then spontaneously shutting down. At that time the phone would only last for 5-10 minutes before the shutdown.
We also tried a Reset of the phone from the iPhone: Settings-->General-->Reset (very bottom)
That didn't work.
Recovery Mode Factory Install:
Finally when I did the final Recovery Mode Factory Install.
http://www.imore.com/how-to-iphone-ipad-recovery-mode
This cleared out the phone and did a fresh install of 10.2
Result - The iPhone 5 stayed charged for 30+ hours and did not spontaneously shutdown
1. after the Factory Install to 10.2 the phone was charged to 100% and left a couple more hours on the charger.
2. the battery showed a rapid % decline at first, but then started to go down more gradually. (over an 10 hour period)
3. at 9% I left the phone overnight and it was at 5% in the morning. (6-7more hours)
4. over the rest of the day the phone went from 5% down to 1% (~ 12 hours)
5. the phone stayed at 1% for another hour or 2 and didn't seem to shut off even though I turned the screen on and kept it onto monitor progress.
6. finally the phone shut down well north of 30 hours.
The battery indication seemed to be out of sync with what has going on with battery.
Battery Recalibration:
I did a battery recalibration as outlined below:
https://www.iskysoft.com/data-recovery-tips/reset-iphone-battery.html
The phone seemed to perform worse after going through the recalibration process.
Then I shut the phone off for a couple of days and left it completely off.
(It seems that shutting the phone completely down is a good idea from time to time.)
I ran a couple more charge and discharge cycles.
Now the phone seems to accurately indicate the battery power.
Last night, I fully charged the phone and left it overnight and the battery indicator went from 100% down to 72% over 12 hours.
I will continue to leave it in standby and see how long it lasts, but the phone seems to be back to where it was before the 10.1.1 update.
The battery readings seem more accurate for a 4 year old phone with an old battery.
Results up to this point:
The battery indicator no longer is falsely showing rapid battery discharges at any time.
The phone no longer spontaneously shuts itself off within 5-10 minutes or the 1/2-1 hr as happened when the 10.1.1 update was installed.
It seems the 10.1.1 update somehow corrupted something and the 10.2 update with a recovery mode factory install followed by battery calibration has resulted in a fully functional iPhone 5 in my case. I know of others who have had success with this method as well.
Hopefully others will be able to achieve similar results.