After lots of feedback from the forums on which this topic was started, I decided to try a fourth method using much finer sandpaper and Mothers Mag & Aluminum Polish instead of Flitz.

At my local Canadian Tire I found 1000, 1500, 2000 and 3000 grit sandpaper and the Mothers polish. With a pair of Nitrile gloves, some soapy water and some polishing towels I was ready to go.

After the 1000 grit:



After the 1500:



The 2000:



The 3000: