There are $\tbinom{2N}N$ ways to choose the hat colors, so it's enough to show that there are $N!^2$ ways for Henry to create the dancing circles after assigning the hat colors. To see that, the $N$ white-hatted people all need to choose a black-hatted person to be on their left; this must constitute a permutation, so there are $N!$ ways to do so. Similarly, there are $N!$ ways for the black-hatted people to choose the white-hatted people on their left. Each of these $N!^2$ choices gives a unique valid arrangement and there are no other valid arrangements, so the number of valid arrangements is $\tbinom{2N}N \cdot N!^2 = (2N)!$, as desired.