I found the same problem, the home button stopped to work. The problem is related to the connectors of the home button, there are two on the left side of the button behind the touch screen and two on the base of the iPhone. When you close the iPhone, after the replacement of the battery, the connector is so delicate that it broken! Maybe is very low quality (!) (I'm very disappointed with Apple!).
Since I don't find the name of this flex cable (it's not the home button but the opposite one where you plug it) to buy it, I found a workaround that consists in to weld 2 wires in order to close the circuit when pressing the home button. Very bad solution but it works!!!