Any soap or detergent that is 'caustic based' will DISSOLVE mildew - TSP, Lye Soap, detergents containing sodium silicates, etc.
Bleach only kills the mildew and leaves the cellular debris behind ... which can become the nutrient source for other species.
When cleaning mildews, ALWAYS wet it out thoroughly so that you dont spread/release the spores. Wear a good high efficiency dust mask or respirator.