Smelling the peppermint oil may be okay. The danger of putting anything in your mouth is that it stimulates the tastebuds which signals to the brain to start the digestion process. That in turn sends a message to the heart to divert energy and blood to the digestion system taking that energy away from the fasting process.
I have heard that fasting can help with arthritis. Chris will know more about that and has probably posted already. I myself have had the experience of having stretch marks healed almost without trace while fasting. The stretch marks I had were on my belly and under my upper arms and were as bad as my first wife's were after our first two children. The body takes the opportunity while not eating to use the freed up energy to do maintenance tasks. Common tasks performed are repairing old scar tissue, relining parts of the digestion system, from the mouth and throat right down to the bowels. Quite often during a long term fast there will be no weight loss for several days, this is when the body is using up the energy from the fat cells and protein from cysts/tumours etc to rebuild old tissue where needed.
The body produces ketones for energy from our fat cells once our blood Sugar drops below a certain level and then when the blood Sugar is depleted our body starts to produce enough ketones to provide full nourishment for itself. That's what we survive on for the duration of the fast, conserving what minerals and nutrients we can't create ourselves.