check the type of "brillo" pad you are using. Don't use an aluminum oxide pad, that will scratch it. You would want a regular type fine nylon pad or fine steel wool.
if you have rust or stains then you could try coca-cola as a cleaning agent- the phosphoric acid will do the cleaning, otherwise look for a chrome cleaner which will be acidic (phosphoric or hydrochloric usually).
metal wadding polish usually works ok, such as never-dull, but to get chrome really nice you need a buffing wheel like above with a chrome polish. and be careful with what polishes you use, chrome is only a thousandth of an inch thick and with the wrong polish, such as for aluminum, you can buff through the chrome when using a wheel. And you want a soft cotton type of wheel, do not use a sisal wheel or anything abrasive.