I've been watching this track for awhile and was having the same issues. I went after the problem in a step-by-step way.
First, the nano. Firmware 1.03 was my problem. I was shutting down within a minute of ending a run, and sometimes just at random. I saw v1.04 and hoped that would correct the issue. And, for me, it did not. So, I went back to v1.02 and now the nano works perfectly. Again, my experience. I did the downgrade myself after finding the firmware on an apple focused website.
I also own an iPhone 5 that I started to use. But, when I upgraded to ios9, I didn't know that the Built-in Nike+ app was removed. And, for those who suggest Nike Running..... doesn't do the same thing, especially on a treadmill. So, if you have a iPhone/Touch 4S->5S, don;t upgrade to ios9, if you want the Nike+ App (again, not Nile Running - no sensor data).
So, now I have an iPhone that I can't use for treadmilling, so I took a look around and finally bought an Adidas Speed_cell. It is a BTLE stride sensor that works like Nike+ fob. Works great with Adidias software, and for Nike fans, use the Wahoo running software, takes to the Speed_Cell and transfer to Nike+.
That is my experience. I wanted to share it.