There can be some confusion in defining the sample variance ... 1/n vs 1/(n-1). The OP here is, I take it, using the sample variance with 1/(n-1) ... namely the unbiased estimator of the population variance, otherwise known as the second h-statistic:
h2 = HStatistic[2][[2]]
These sorts of problems can now be solved by computer. Here is the solution using the mathStatica add-on to Mathematica. In particular, we seek the Var[h2], where the variance is just the 2nd central moment, and express the answer in terms of central moments of the population:
CentralMomentToCentral[2, h2]

We could just as easily find, say, the 4th central moment of the sample variance, as:
CentralMomentToCentral[4, h2]
