Ouch. Valuable lesson learned, I hope? ;-)
That stuff is fine particulate; it can stay in the air for a while. How long depends on a few things, but mainly: exactly what kind of fire extinguisher was it? Are there air currents in the house disturbing it? Did fireplace ash get all in the air as well? (BTW, your flue was CLOSED, not open. I always start by lighting a single piece of paper in the fireplace; this will both indicate if the flue needs to be opened AND get a draft going up the chimney once it is opened.)
The thing that occurs to me though (personal experience here) is that if you use a vacuum cleaner to deal with the mess, and it doesn't have a *clean, fully functional* HEPA filter on it, then your cleaning efforts will actually put that stuff back in the air, even more effectively than the fire extinguisher did.
Since it can't be good to breathe that chemical, please wear a dust mask, respirator, or at least a wet bandanna or something over your face, if you are going to be exposed. And use this website to look up the MSDS for your fire extinguisher; the Material Safety Data Sheet will not only tell you WHAT is hanging in your air, but it will give tips on how to clean it up. Good luck!