If I try to give them something and they have a fit about it I assume they know why they don't want it.
I usually find out they were right.
If you just hold the cat on you lap and direct the nebulizer across the front of its nose it will breath in the silver. If the cat revolts try again later. Do not direct the nebulizer directly into the face. The cat will freak and I don't think silver is good for the eyes.
I no longer give my pets vaccines past their initial series. Too many bad reactions. This could be the cause of the cough.
Some balanced supplements and continue with the oral silver sounds more reasonable. Cats love brewers yeast. Full of B vitamins and protein.
Don't give to animals with kidney impairment as it raises uric acid levels. Just a teaspoon twice a day. Too much will make them sick.