Touch ID and Battery problem after upgraded to iOS 9.1

First, reset the phone by holding the sleep and home button for about 10sec, until the Apple logo comes back again. You will not lose any data by resetting, but it can cure some glitches.

If this does not work, and to make sure that this is not software related, set it back to factory settings, without using any backup data afterwards. Set up the rest of the personal settings manually and test the phone.

If the phone works as before the update, restore from the latest backup. If the problem comes back, the backup data contained corrupt data, causing your issues.

If it still does not work, this could be a hardware issue and the phone has to be serviced by Apple or an Authorized Apple Service Provider:

Use iTunes to restore your iOS device to factory settings - Apple Support

Find an Apple Authorized Service Provider

iPhone - Contact Support - Apple Support

Sorry that it did not work, you should get the phone checked by visiting an Apple Store or an Apple Authorized Service Provider.