To get a mirror finish you will need to use a buffing wheel and polishing rouge. Scotchbrite will give you a brushed finish. Or you can buff it by hand with the rouge. Your arm may get sore after a while though
.
NEVER use steel wool to rub on a stainless steel
. This will embed small particles of steel into the stainless and will cause rust spots. Bronze wool would be better but it will not give you the mirror type finish you are looking for.