Don't be afraid to 'muscle' the polish a bit. Flitz's polishing grit is very fine (about 3µ); as long as you're using a CLEAN rag/cloth, the polish won't leave any coarse scratches, and should only produce a high shine. Dirty rags used to polish may have gritty content on them, which might scratch (likely will, in fact).
Another idea: May sound strange, but re-application of some of the vinegar might help remove some of the stain (try it on a Q-tip). Just be sure to immediately wipe it off and completely wash it, to make sure there's no trace acidity left behind. If that helps take most of the staining away, try the Flitz again to finish it up.
If that doesn't quite get it done, some very light fingertip sanding with some high-grit wet/dry sandpaper should quickly remove the staining. Try some 2000-grit or higher first, which will produce a pretty high shine, perhaps with some very fine 'satin' scratches; then follow that with the Flitz polish again, to clean that up and bring the mirror back.
David