It's even worse with non-anodized aluminum, like mine. Skin oils accelerate the oxidation, leaving a black oily grime on it all the time. The mail-patterns left on my forehead after I take the coif off are amusing... I call them "tire tracks".
With my aluminum, I've only cleaned it a couple of times, but liquid dish soap and a good rinse do a very good job of it. Since it's not steel, I don't have to worry about rust here.
For removing rust, as Wolf said, putting it in a bag with something abrasive, and sloshing it around, will work. Some people use sand, others use fresh kitty-litter. I
think it was Jeff Hedgecock of Historic Enterprises that I saw on another forum who recommended not using any sort of sand, but rather wood chips like you'd use in a cage for rodent pets. He said it's enough to scrape the rust off without ruining the finish, and it'll absorb the rust powder (and the rings themselves do most of the work of scraping the rust off).
But your mail is brass plated right? The ultrasonic cleaner sounds like a cool idea. You could probably get away with the soap method I discussed too... I just stuff the whole thing in the sink and get a whole lot of the soap on my hands with a little water and go to town on it.
I've found "Brasso" to be
really good at removing brass tarnish, but it might be awkward to try to use it on mail.