This is how I do it. My daughter plays flute and has this cloth lying around.
I guess it's the same as jewelry type cloths, but I feel good that it"s made for musical instruments.
One day I tried it on the hardware of my guitar, it works great!
I bought my own and she can have hers. By the way, it cleans up nickel strings pretty well too!
It also makes the gold trem on my CU 24 shine. I wish I tried it a long time ago.