Looks like iOS 9.1 is causing more trouble than good.
People are reporting that due to the new software update, TouchID forgets their fingerprint or that the sensor does not work properly. Apple's support forums are full of people claiming issues on a lot of apple devices. If you're experiencing the issue yourself, you'll just have to wait on a fix.
For now, Apple is aware of the problem and is planning to solve it somehow. We have to be pacient.