Due to the special bond between Touch ID and the Apple A7 CPU inside the iPhone 5s you will have to continue using your old home button. Any replacement button would remove all fingerprint functionality from the phone.
Therefore, you have to transfer your home button into the new screen. To do this, you need to loosen this screw first.
Bend the cable out of your way a little, remove the following two screws and the metal cover they secure.
Delicately scrape off the cable starting from the top right corner using the plastic spatula. It is pretty tough, but be very careful nonetheless! If it is damaged, Touch ID will not work on this phone.
Take care not to damage this little loop, as it is responsible for holding the home button cable in place.
Gently press against the home button from the front of the screen towards the rear until it starts to come off. Now you can remove the whole package — button, cables and all.
Grab your replacement screen and put the home button in the same place from where you removed it earlier. Match up the loops with the little nubs on the rear side of the screen. When everything fits nicely, gently press down on the rubber part around the actual button with the plastic spatula.
Put the metal cover back and secure it with two screws.
Gently press the cable back in place so it sticks to the back of the screen. Fold it back down and secure it with the remaining screw.