Moments from tail bounds via the layer-cake formula

Suppose $X\ge0$ satisfies $\P(X\ge t)\le 2e^{-t^2/K^2}$ for all $t\ge0$. Using the tail formula $\E[X^p] = p\int_0^\infty t^{p-1}\P(X>t)\,dt$, prove that

$$\E[X^p]^{1/p} \le CK\sqrt p$$

for an absolute constant $C$ and all $p \ge 1$. Compute $\E X$ and $\E X^2$ explicitly in terms of $K$.