Douse a cotton ball in rubbing alcohol. Don't merely dampen it; get it wet enough that it will drip if you squeeze it moderately. Then rest it on the DS and squeeze it enough to release some of the alcohol onto the dirt. Hold it there long enough so that you moisten as much of the dirt as possible. That step is the most important! Then rub vigorously.
I have no idea that this will actually work, but it provides two things that might help: The alcohol might be able to soften up the dirt, and the fibers of the cotton ball might be able to penetrate into the tight space.
If alcohol doesn't work, try the same trick with plain old water. I'm not sure what kind of "dirt" is in your DS, but water is a polar solvent and rubbing alcohol is a nonpolar one, so you'll have you bases covered on the cheap.
