We need a Sticky post made for this subject!
First we need to see the pictures..
That way we'll know if you need to simply clean and buff it, or need to transfer it to one of us that has experience on an overly dirty stainless wheelgun.
Or, we might be able to advise you on how to do it the easy way. I'm just not sure which it is yet. It's really hard to mess up a stainless handgun if you're reasonably careful not to burnish off the sharp edges.
Here is a link to a recent thread that cover the subject pretty well.
Try Mothers Mag Polish. I bought some MAAS silver polish for mine and a soft polishing cloth.
Just about any high quality Aluminum or Stainless polish will do wonders. An old CLEAN worn tee-shirt works very well, without adding more scratches from the rubbing. It's really very easy, but time consuming.
If you were in the military, it's much like polishing brass. Not much fun to do it, but it'll make everybody smile with pride once it's finished. If not, it's really not a lot of work, but it take time and patience to get it evenly polished across all the surfaces.
I've seen Stainless revolvers that reflect an image better than some mirrors.
To remove any Lead deposits and gunpowder residue I'd recommend a Birchwood Casey Lead removal cloth, or another brand that serves the same purpose. Dremels are too small and will usually leave streaks if you aren't ultra careful. You can use a drill press and Jewelers Rouge if the speed is set slow enough.
We have a couple of really good football games coming up in the next two weeks, so you'll have a perfect excuse for some couch time. Just tell your mate that you need to sit down and concentrate on the polishing and the TV helps relieve the intense focus required to do it right.
I'm sure others will chime in shortly with some additional tips and help.