Are you using a machine to polish it out? What product(s) are you using?
A machine will be a lot faster and more effective at bringing back the gloss. You're probably going to be best served with a 2 step process of a more aggressive polish/compound followed up by a fine finishing polish/compound.
I've been very happy with Rupes products, which in this case would be their DA Coarse & DA Fine Compounds. I use those with the matching foam pads from Rupes. Microfiber pads (also matching the coarse/fine designation of the compound) will speed up the process. For a mirror cap, I use a mini polisher with 2 or 3 inch pads for better control on the small surface area.
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