any detergent will act as a surfactant, a surfactant acts to lower the surface tension of a liquid or that between the liquid and the brass. To remove the patina you will need a mild acid of some type. Powdered or natural citric acid, vinegar etc. I did some research on the lemishine, most analysis shows the active ingredient be just citric aid
here is a long read on lemishine for those interested
http://www.chemistry-blog.com/2012/0...in-lemi-shine/
pure powdered citric acid is available pretty cheap if you want to save a dollar or two. Like a previous poster said a pinch goes a long way so one package of whatever should last you for years and years