Go to the car department and get some chrome cleaner and wax. I have a towel stand in my bathroom that was pitted and looked awful from the moisture in the bathroom. A treatment with the chrome cleaner and an application of car wax and it looked good as new.

Or try Bartenders Helper.

Are you sure it is real chrome or is it chrome looking plastic?

Be careful if isn't real chrome. Otherwise I go for Barkeepers Freind. Best product for so many cleaning jobs in the kitchen snd bath.