EDIT: I suspect I may be going about this all wrong, so maybe disregard this.
So, I can get this far on my own: $E(\hat{\sigma}^2) = E(\frac{1}{n - 1}\sum_{i = 1}^n (X_i - \mu)^2) = \frac{1}{n - 1}[E(\sum_{i = 1}^nX_i^2) - 2n\mu E(\bar{X}) + n\mu^2] = \frac{1}{n - 1}[E(\sum_{i = 1}^nX_i^2) - n\mu^2]$.
I know that $E(\bar{X}) = \mu$ and I'm given a formula for the variance of $\bar{X}$, so I suspect I'm somehow supposed to get from $E(\sum_{i = 1}^nX_i^2)$ to $E(\bar{X}^2)$, but I don't see how. I also have a formula for the covariance between each $X_i$ and $X_j$, though I don't see how I would use that. I don't know the distribution of the population I'm drawing from, so I can't evaluate $E(X_i^2)$, right?
Thanks in advance.