My cat is asthmatic and sometimes he requires the prednisolone daily also, usually in the winter time when the furnace is running.
Things that you can do to help: Ask the vet to increase the dosage of the inhaled steroids, and try administering twice per day rather than the once prescribed. Put filters over the furnace vents in each room, replace the furnace filter with a HEPA filter, run an air purifier, replace carpet with flooring if possible, have the furnace vents cleaned, switch to unscented litter and laundry detergent and consider that it could be food related and try some different foods with fewer ingredients.