First, you need to clean all traces of visable mold. You can do this with bleach or hydrogen peroxide. I would personally use the latter because it is less toxic. It is also important to make sure there are still no leaks or it will just come back. Active mold sends literally trillions of spores into the air which settle in carpeting, all nooks and cranies, pourous materials like raw wood, etc and when conditions are right, you get problems. It may be a good thing to have the variety of mold tested because some molds are very, very toxic, even in spore form. If you are dealing with one of these it can cost 1000's of dollars for abatement. These types of molds are treated with extreme caution because they can cause serious and permanent damage to many body systems. If it was me and I knew the problem wasn't one of the carcinogenic molds I would still be very, very careful.
I would certainly change the ductwork( I wouldn't try to clean it)seeing as it was found in the heating system. All you have to miss is a few spores and when conditions are right your back where you started. Mold in the ventilation system is particularly insidious because it distributes spores throughout the house.
How is the basement. If you can smell mold then you have an immense problem. It has likely permeated the structure. I have a good friend who is one of the top indoor environmental experts at a prominent University with a particular interest in molds. His advice would be to proceed with extreme caution because of what mold can do to the human body. He would likely say if its a significant problem, do not move in. He has tried to help many people in this situation and finds if it is a significant problem that it actually costs more to try to fix the problem than it does to rip down the structure and start over. He finds that most people don't give mold the respect it deserves and then find that their life is turned upside down as a result.
If the mold is indeed isolated then you must clean all visable mold, utilize true hepa filtration(not cheap department store models)and do a series of high powered ozone shock treatments. Need a high powered machine for this which can produce about 5000 micrograms/litre. Do several treatments for a few hours at a time. Do not enter when treatment is underway or until ozone has dissapated. This can take several hours after the treatment. Also when cleaning up utilize hepa filtration respirators and gloves. Discard clothing when done.
Do not move any personal belongings in until you are sure the mold is no longer a problem otherwise you can contaminate these items and face throwing them away as well.Keep the problam isolated.
Then run your hepa after treatments are done to pick up stray spores. Then you would make sure that you have good ventilation out of your kithcen, laundry room, and all bathrooms. It is important to not let humidity build up in your home or it will all start over again.
Just look on the internet to find the serious stories of how mold has impacted many people's lives. This is a situation that must use extreem caution when dealing with.
Sometimes releastate documents have a clause for mold so you may have some recourse here. If you do, walking away may be the best thing. Sorry to say that but I know too many people who have ruined lives because they didn't think mold could be that serious. Your health is not worth gambling on. If you have chidren then its a no brainer.