You can buy spray bleach, which i used all over the shower, left it overnight, sprayed it off and all mildew was gone and the tile grout was far better, almost back to whie. some months later black stuff appeared again, so this time I used some of my ordinary bleach from a asda cheapo bottle, put it into the spray bottle with some water, sprayed again, and next day all white and lovely.
It really does remove the black mould, our shower never looked so good.