you only need 3 things to get touch id to work
1. the orignal home button that the phone came with
2.a good cable (via home button and cable running from mobo to home button.)
3. good chip on mother board (because home button is paired with main chip from factory)
i have seen the cables go bad most often, even new ones can be bad from factory, or manufacture, ie china's poor quality control. so i would try a couple new cables that run from the mother board to home button flex.
also check home button as well some times the plastic connector breaks on the inside and will cause that issue too, but if you use the new itunes with new os download you should not get error 53 unless chip failure cause they have removed error 53 on new ios versions do to class action lawsuits. thanks apple lmao
also if you have iphone 6s the cable is built into back-light housing and will cause this as well, only thing you can do on that is get another replacement screen, or swap back-light housings but that repair is expert only, mainly cause that repair is usually done in a clean room. best of luck