Use a little kerosene or WD-40 with that wire brush if you are doing it by hand. Seafoam or B12 Chemtool is another option, but would be smarter to use if you take the cylinder off. Those solvents are good about breaking up carbon and other grit and will also act as a lubricant to help prevent scratching anything too badly. Just wipe everything up well with some towels and make sure everything is super clean before putting it back together, and I personally suggest wiping a little 2 stroke oil on the inside of the cylinder before re-assembly as well.
A little ash and carbon buildup is a normal thing for a lot of 2 strokes. If you are running on the rich side of things, which you are from what I can see, it will be worse.
One thing you can do to help reduce that was already suggested, which is running an ashless oil. I have been using VP Racing synthetic 2 stroke oil for a few years in all of my 2T yard equipment and motorized bike builds. It's a very clean running synthetic, and it's also easy to find at many retailers like WalMart, ACE, etc.