Ice cubes. Salt. Shake/spin. Dump. Repeat if needed. Rinse when finished.
I worked in a restaurant for the better half of my teens and twenties... We used to do a sanitizer soak on the glass coffee pots 2-3x a week, but on the other nights we would use ice cubes and 2-3 tablespoons of table salt and then vigorously "spin" or aggitate the glass cannisters to clean them. The salt would act as a mild abrasive, and also slowly melt away the ice cubes creating a slush wash to scrub/rinse the walls of the coffee pots. It worked like a charm on glass and im sure it may work wonders on the stainless as well.