Good question, Dave!
I doubt that you could ever polish out a "satin nickel" finish to a "bright nickel" finish. It just won't happen.
I have been led to believe, here on this board, that with patience and an amount of effort you can, using a product such as "Birchwood Casey Lead Remover & Polishing Cloth," turn that satin nickel finish into something that more resembles an antique nickel appearance. I have seen the results and, to my eye, it is much more appealing.
Hopefully Yahoody, our resident "expert" in such matters, will join in and give you more specific advice.
CJF