This is a difficult problem, especially since estimating the volatility faces a number of issues:
- the classic "pollution" of realized variance by bid-ask bounce when using intraday data (cf Aït-Sahalia, Mykland, and Zhang);
- including overnight gap effects if using daily or less frequent data;
- volatility changing (hence the utility of GARCH and related models); and,
- the possibility of structural breaks (cf Timmerman) or jumps in the mean return and volatility (cf Todorov and Tauchen).
You also need to determine what is a reasonable way to estimate $V$ since that is usually an average. (Hence why the $V$ is usually written as $\bar{V}$.)
That said, estimating price impact is imprecise, so to some extent these concerns are less important than the uncertainty of estimation.
In my experience, price impact modeling quickly gets into "secret sauce" which means getting a specific answer about estimation is difficult to impossible. What I will say (i.e. what I will allow myself to say) is to try a recent estimate of $\sigma$ that it stable and to use a similar estimation period for your average volume $\bar{V}$.
Finally, I would be remiss if I did not suggest you consider other price impact models. I mention a few better models here.