I was reading the Taylor 2004b (instruction book that comes with K-2006 test kit) and it said that chlorine will not eliminate or kill black algae. The manger/owner(?) of the pool supply store in which I was reading this, echoed those same comments. So of course I bought some Black Algae Algaecide.
Additionally, he said to be sure to vacuum to waste any area where you have brushed.