Apply a small amount of chrome cleaner (available in automotive stores) to the surface of the sink. Use a gray course nylon-scrubbing pad and scrub the sink in the direction of the grain. Add more polish until you have coated the entire surface of the sink. Continue to rub the pad in the direction of the grain, adding more water if the polish dries out.
Rinse the sink with warm water and wipe it with a microfiber cloth to wash away any of the grit remaining from scrubbing the surface of the stainless steel.
Add more chrome polish and buff with a maroon nylon-scrubbing pad, still in the direction of the grain. Use the corner of the scrubbing pad to work the corners of the sink, the drain area and around the faucets. You should be able to see some of the shine coming through the haze of the polish.
Rinse the sink with warm water and wipe the surface with a microfiber cloth. Buff dry with a dry cotton towel until no haze remains.