Looks like this paper could help you: Evaluating Kolmogorov's Distribution
Check the 3rd header, Limiting Forms, for a mention of how the moments are found.
Not only do all moments exist, but they are all simply expressible in analytic form:
$\left<x^m\right> = \frac{\Gamma(m/2 + 1) \, \eta(m)}{2^{m/2 - 1}} $
You can obtain this formula by using the form of the series definition containing factors $e^{-2 k^2 x^2}$ and integrating term-by-term.
(Sorry to have to add this information via edit, but "protecting" this question from answers by people who have not used the site previously prevented me from putting this in a separate answer.)
