Sorry for the inconvenience you are experiencing with your iPhone 6.
This device has a fingerprint sensor that is very sensitive to ANY work that might be performed to replace a broken screen. The cables\leads that run from the button and behind the screen can be damaged very easily. This damage can prevent your device from being able to use the sensor. Great care should be taken when replacing the screen.
I recently had my device repaired, but was fortunate to be able to keep the original home button. This is important. Apple has designed the iPhone 6, 6S, and 6S Plus with a feature that can disable the fingerprint sensor and cause an Error 53 warning to appear when attempting to update to the latest firmware. Apple has recently corrected this by reissuing firmware that allows the user to use the home button but without the sensor.
The good news is that if your device is not under warranty Apple will be able to replace your entire screen assembly for the same price as a third party vendor but you will receive genuine parts and Apple will calibrate it for you on site. Granted you will need to make an appointment with the Genius Bar and be willing to sacrifice a few hours of your time. But in the end, if no other problems exist with your phone, you should have a new screen and home button.
REMEMBER: Apple will ask you to disable Find my iPhone when you bring the device in. They will also have you disable the lock screen passcode so they can have access to the device while it is being repaired. Remember to perform a backup before you go in.
Here is the latest pricing chart from Apple long with the link.
With AppleCare+ Without AppleCare+
iPhone 6s Plus $ 99 $ 149
iPhone 6s $ 99 $ 129
iPhone 6 Plus $ 79 $ 129
iPhone 6 $ 79 $ 109
iPhone 5s, iPhone 5c $ 79 $ 129
iPhone 5 $ 49 $ 129
I would love to hear how this turns out for you.