I'm having the same issue as you described it here. I have a iPhone 5, too.
If you have done anything suggested by Apple, it's a battery issue. You need to change the battery in order it to work. Otherwise, I don't think there is any other way you can make it work. Attached is the link from Apple.
For my situation, before it turned out to error 2001, the temperature of my phone was pretty hot, so I shut it down to let it cool. But after a while, when I wanted to reboot it, it turned out like this.
So I had been searching around the internet for over a hour to seek a solution, all the people were suggesting to try the fix by above link. For me, it did not work. After having reading several forums, they suggested to change the battery. Therefore, I decided to change my battery.
At the same time, I didn't really want to take it to Apple Stores because I think my problem lies on the battery and it would cost me over 70 dollars to let them fix it.
So I ordered the iPhone 5 battery tool kit from Amazon. It was around 15 dollars. I followed a tutorial on the internet to change the battery. It's pretty easy. Then I fixed the problem.