Here is what happened to me, and it may have happened to you as well. I replaced the battery in the iPhone 5s, which is a bit more difficult than the 5 or the 4s, but as I had read the instructions repeatedly, I knew to be very careful about the cable for the Home button. No prying damage, nothing. When I put the iPhone back together the Home/touch button stopped working. Similar to most complaints. I ordered and received a new Home button after disassembling and reassembling iPhone repeatedly. When I had put the new button in I wasn't certain about how the cable should lay in. I drew it across the button so that it was folded immediately after the connector, but then it layed flat across the metal Home button cover. It works fine now. I inspected the old (original) Home cable and found that it had another fold, one that I would not have recognized if I had not had to install a new button. I noticed that there is a small contact on one side of the Home button cable, and the cable had folded in such a way that the contact was not in contact with the metal Home button cover! Such an additional fold probably happened as the unit was closed because I didn't know to preclude the possibility of it folding at any particular point. There is no apparent damage to the cable. Try this, it is very likely the problem we encounter with the battery install.
Best always,
Drew