Cato wrote: Using a wire brush and the powder should do the job!
You should not use a wire brush with a SS chimney. It is recommended to use a poly chimney brush.
I have a prefab SS chimney and at the end of the year I use my poly brush to clean all ash off of the sides and then I made a sponge between 2 pieces of plywood (made to fit chimney) that goes on my rods and I wash the inside of the chimney down with a baking soda and water mixture. I don't know if it helps or not because this is only the second year for the chimney.