If my wife is not fasting then I would ask her why, if it is medical reason than I would encourage her to see a doctor and see if we can solve the issue find a solution.
If it is non-medical than I would ask her to keep at least 1 fast in the whole month. After she has completed her first fast, I would encourage her to keep another fast. I won't push her, just encourage her like setting a milestone in ones life and trying to work towards it.
IMO the best way to tell someone what Islam is or what being a Muslim is, is to show them by actions. I used to ask my non-muslim friends to fast with me once in ramadan if they wanted to know how I feel during fast, most of them did try and they got to know a bit better about Islam I guess. I also used to take them to mosque if they wanted to just sit in the back to observe what we were doing, during jumma or during eid namaz or during halaqas. We have to be open towards them, invite them to experience our lifestyle and then inshaAllah rest can be left to Allah to put noor in their hearts.
I have gone to churchs, sit through their sermons just to show my willingness to learn their religion and in return they learn a little about my religion. The same should be done with the wife (in this case). Get her indulged in fasting slowly. Like we do with our young children. We ask them to start by keep the last 10 fasts and then so on.. they might not understand completely about why are they fast except for the "coolness" factor but I guess its OK to feel "cool" to be a Muslim
spread the religion by love, not hate.