Using a non abrasive cleaner and a soft cloth such as washing up liquid wash the chimney. Make sure the water is hand hot. Thoroughly clean the chimney with the soapy water changing water if necessary. Dry the chimney with a dry soft cloth. Then with a furniture polish buff the chimney to give it a shine.
I would suggest using a cloth with warm water and then dry it trying not to leave any water spots (those can be annoying! ). There are glass cleaners that can be effective if you have fingerprints that need to be removed. You can also search for a stainless steel cleaner, which should get the job done.
You can use 6" round metal brush and poly pro rods for cleaning your chimney. But I am afraid that it may need lot of muscle power to use this brush and rod. If it is not a burden then you can contact the cleaning agencies available to help. Why should you take the risk which is not safe.
Just like you would clean any other chimney with the regular chimney brush thats if you want to clean the inside of the chimney. If you are trying to clean the outside of the chimney and it's stainless steel us scotchbrite or a flapper wheel on a grinder those are the only ways that will work and not ruin stainless.