Complete stainless steel kitchen in our rented house - I was getting really frustrated keeping the stainless steel worksurface clean (not to mention had a heartattack after dh used a scratchy sponge all over it - shudder). The only thing I found was some horrible stinky white liquid to polish it with - what a drag that was. One day I was complaining to the elderly neighbour and she got up went home and came back with a couple of cloths - yes all you need is a microfiber cloth!! They come in different weights and you just wet it slightly, wring it out well and off you go - no detergent, no polish nothing except this cloth which shines and cleans everything: stainless steel work surface, kettle, door handles, taps, windows and mirrors are a dream to do (they might get done more than once a year now).
I'm sure you'd get these clothes (not cheap but worth it) at the hardware store or supermarket cleaning aisle. They are like a giant reading glasses cleaning cloth.