Bilberry is the best wheel cleaner that I have used so far. It works well even when diluted.
To make life easier in future I would recommend a deep clean and sealant, just about all of the brake dust will then wash off with warm water, car shampoo and wash mitt or sponge.
A fallout remover such as Iron-X will dissolve 99.9% of the brake dust ingrained in the wheel's finish coat. Smells rank though.
A clay bar will pull out the other 0.1%.
Use a light polish to restore the finish coat condition.
Apply a proprietary wheel sealant or synthetic sealer to stop brake dusk becoming ingrained. I used Chemical Guys Wheel Guard and was very pleased with the protection it afforded. Now using Finish Kare 1000 synthetic paste wax. The CG Wheel Guard is much easier to apply and buff off compared to the FK 1000.
I use this regime on my winter and summer wheels once per year. For the rest of the year I just use warm water and car shampoo.