I had this problem too on my iPhone 5s.
I first replaced my screen (touch id was working after replacement) I updated my phone from ios 9.3.2 to ios 10.0 and then touch id stopped working
What I did was downgrade it to ios 9.3.5 while it was still availible and touch id came back.
I updated it recently to ios 10 and guess what? Touch id can not be activated on this iphone. Sucks :/
I guess they added new security on ios 10.
Your best option unfortunantley is paying apple the 80$-90$ for a new touch id.