You shouldn't worry about skew or kurtosis in your Kelly calculation. In fact you should probably use fractional Kelly and just stick with a normality assumption.
Really? You want derivation? Here is one. https://sites.math.washington.edu/~morrow/336_10/papers/jane.pdf If you want to include skew and kurtosis, just replace the Gaussian with your own favorite distribution that has skew and kurtosis. But don't ask me because I am math challenged and only care about the answer.
Do any of you actually have practical success with Kelly? R Vince says it is not the right model for trading and optimal if is better for trading. I would be interested in your thoughts...