What can I do to help my cat with asthma?

The only thing I would add is a humidifier in case the air is too dry. I use a cool mist one. Air purifiers seem to be trickier to choose than they should be. Some only act as dust filters and some claim they do more things to your air than seems possible. I use a purifier with a washable permanent filter - sort of like that Oreck thing they advertise on TV. It certainly catches dust and cat hair but whether it does more - who knows? My Siamese had asthma but not as severe as your cat has. We did the prednisone bit but were never really satisfied with it either. Good luck.