Electropolishing is the way to go if you want mirror reflection.
Polishing stainless steel (or any other metal) is a process of progressive mechanical abrasion.
The goal of the abrasion (polishing) is to cut finer and finer scratches into the surface until they are invisble to the naked eye.
"Electro polishing is a selective electrochemical metal removal process, exactly the reverse of plating. Simply stated, the electropolishing process results in the simultaneuos smoothing, leveling, and brightening of stainless steel material by adding low-voltage electric current to a flowing chemical solution.
When the proper combination of electrolyte, current/voltage temperature relationship is attained, the high points of the surface irregularities, or high current density areas, are selectively removed at a greater rate than the remainder of the surface.
Electropolishing selectively removes surface flaws and embedded impurities, as well as high points in the surface layer.
An electropolished surface is essentially equipotentialized. Local corrosive cells cannot be set up, because galvanic differences due to stress in metal surfaces have been eliminated.
Electropolishing passivates stainless steel to a greater extent than any other process. It applies an invisible protective oxide fill on the surface which imparts a condition of maximum resistance to attack and corrosion.
With surface impurities removed, and a level, smooth surface that has been brightened by electropolishing, the coefficient of friction is greatly lowered. A smooth surface eliminates turbulence and the erosive action which generates particulates. The bright smooth surface is bacteriologically cleaner than any known method of cleaning or polishing."
http://www.southernstainless.com.au/index.php?page=electro-polishing 
