How was Kakashi Hatake able to survive and become the next Hokage after losing his Sharingan?
Very soon in anime,


  Obito will say that his gift for Kakashi was not full and will give him both Sharingans. He will manage to use Susanoo and save Naruto, Sasuke and Sakura from Kaguya's attack. I don't know why, but after war he has normal eyes, maybe he can deactivate Sharingan.


UPD.

I've read that chapter again and understood why he has normal eyes. 


  Obito says that his present may expire but it's not completely useless.