Ouch! You are between a rock and a very hard place.
The TouchID button has two parts to it the power switch part and the finger print sensor. Sadly you need a new TouchID button! But replacing the button on your own will kill your system as the TouchID part is tied to the logic board logic.
Sadly you have no choice here you need to go to Apple to have them replace the TouchID button as they have the needed programer (which is not available to anyone outside of selected Apple repair depots.
Do you have AppleCare+ service contract? If you do that will help to take the sting of the repair costs.