Flitz Aluminium preclean, Flitz polish and a Flitz polishing ball.
The preclean is pretty aggressive and used when you have serious oxidation.
The Mother's foam balls are kind of crap, they shred and are a one-use kind of thing. The viscose that Flitz uses on their (highly overpriced) balls is much sturdier and reusable multiple times (just wash them in between). So I suppose overpriced is in the eye of the beholder... I don't regret buying them.
Personally I suspect those stains are in the coating, not the metal. The coating on these (and my Touring) is crap. So maybe you should consider taking the coating fully off the wheels and then just polish the bare aluminium. Not sure what would remove the coating... paint stripper?? But no, that would also do away with the black areas...
Powdercoating is the easy way out but it just doesn't look as good as aluminium polished to a mirror. The downside with the polished metal of course is that you have to go over it regularly to keep it shiny.