This is a tough one because it is someone's home. Asking them to accommodate your intolerance for smoke is hard because, as I see it, it would require more than just not smoking while you are there.
I have an extreme sensitivity to this so I know, for myself, I would need it Very clean, everything wiped down, no drapes or other fabric that would hold the stale smell.
I like the idea of letting the client know you have this sensitivity. If they value your care as much as you know they do, perhaps the wife will be willing to do whatever it takes to keep you coming.
Your employer may assume this is a risk you are willing to take & if you are not, may pull you from that assignment as it is not a good fit for you as a caregiver.

I would not be able to tolerate the smoke smell myself, but if a job were in the balance I would try very hard to find a solution with the client. I hope you can find a happy medium that everyone can live with so you can keep this job.