  We study the analytic continuation of Feynman integrals from the kite family,
expressed in terms of elliptic generalisations of (multiple) polylogarithms.
Expressed in this way, the Feynman integrals are functions of two periods of an
elliptic curve. We show that all what is required is just the analytic
continuation of these two periods. We present an explicit formula for the two
periods for all values of $t \in {\mathbb R}$. Furthermore, the nome $q$ of the
elliptic curve satisfies over the complete range in $t$ the inequality $|q|\le
1$, where $|q|=1$ is attained only at the singular points
$t\in\{m^2,9m^2,\infty\}$. This ensures the convergence of the $q$-series
expansion of the $\mathrm{ELi}$-functions and provides a fast and efficient
evaluation of these Feynman integrals.
