For anything I want a mirror polish on I use the following progression of strop compounds. 14,10,5,3.5,1.0,0.5, and 0.25. This may be a little extreme for what you are doing but maybe not?!
For knife edges or flat things I like to strop them with balsa (far better than leather in my experience--way more feedback) but for blades, bolsters, ect I use flexible leather from an old leather jacket with some compound on them. This is a lot of work by hand so sometimes I just a dremal with a polishing tip.....okay this is mainly what I use but the leather is nice for touch up.
This is great if you have these compounds and strops but if you don't I know you can use filtz or even auto polishing compounds. I have never tired to get a mirrorpolish with those before b/c I use the above, but it may work. You could also go to a grocery store and look at the silver or brass polishing products (make sure to check if you can use them for stainless steel). I remember I had to polish my Mom's silver sets as a kid and I def got a mirror finish on those bc I made funny faces in the spoons!!!
Good Luck!