I had a similar problem with rusting on "stainless" steel of a stove top in a previous motor home.  I got tired of polishing and having the rust return, so took the top off (two thumb screws) and looked in the local phone book for a chrome plate shop. (The Harley motorcycle/antique car folks know where to find these chrome shops, they are generally smaller outfits).  They prepped it,
and chromed it.   Talk about a shine! Cleaned up after spills easier than before, too.

Ray in Rochester