So I think I figured out that I should use the distribution of s^2 somehow: I know (n-1)s^2/(sigma^2) is distributed according to a chi-squared distribution of df=n-1. And I can get the MSE(s^2) = E(s^4) - 2E(s^2sigma^2) + E(sigma^4), but I'm not exactly sure how I can use the distribution to calculate E(s^4) and E(s^2sigma^2)......