I did this around August last year when the weather wasn't too hot. It worked very well. I saw a huge improvement in the leaves. The leaves did come back and they looked much healthier this summer and not one spot on any of the leaves.
Neem oil is primarily an insecticide, it does work on powdery mildew, but you will need something much stronger. Mancozeb or Captan both these products are multi-fungal killing fungicides. I understand your problem I am fighting the same battle as you.
