When you try to log into the OnTheGo and the touch ID isnt responding, you'll need to sign in with your password first. Normally there will have been an update that requires your password to enable all the updates. Once you are logged in again it should be working.
However if it doesn't work, then verify that "Enable Touch ID" is turn on in the setting of OnTheGo first. (Lower left, Gears -> Settings -> turn on Touch ID.
If that is on and your touch ID is simply not reading your fingerprint, go back into Ipad Settings (Gear)->left side of the screen find the "Touch ID & Passcode" and enter in a new finger print and change the name to the finger you just used. Delete out the old one.
You will need to close and re-log into the application again.