How precisely can you estimate volatility?

Let $r_1, \dots, r_n$ be iid with mean 0, variance $\sigma^2$ and kurtosis $\kappa = E r^4/\sigma^4$. (a) Show that $\hat\sigma^2 = \frac1n\sum r_t^2$ has variance $(\kappa - 1)\sigma^4/n$, and use the delta method to show $\operatorname{sd}(\hat\sigma)/\sigma \approx \sqrt{(\kappa-1)/(4n)}$. (b) How many daily observations give a 5% relative standard error on $\sigma$ under normality? Under the Nifty sample's excess kurtosis of 15.2? (c) Why is (b) still optimistic for Nifty? (Hint: 2.6.)