I cheated. The year we replaced our steel sink with a fancy porcelain one and I started having to clean those black marks and ruining my beautiful new sink I went out and bought those rubber mats. Took em out after doing the dishes and rinsed and swiped and it looked great! (that is until DH got left at home alone for a long while and dumped his coffee in the sink wihtou taking out the mats. I though I'd have to try that Efferdent stuff because nothing else was working).
Vinegar is great stuff for hard water and lime deposits. Get a gallon bottle or two.