Since triangle ABC is similar to triangle JKL (denoted as \( \triangle ABC \sim \triangle JKL \)), corresponding angles of similar triangles are congruent. Therefore, the measures of corresponding angles will be equal.
Given that \( m \angle A = 65^\circ \), \( m \angle B = 13^\circ \), and \( m \angle C = 102^\circ \), we can determine the angles of triangle JKL.
Since \( \angle K \) corresponds to \( \angle A \), we have:
\[ m \angle K = m \angle A = 65^\circ \]
Thus, the measure of \( \angle K \) is \( 65^\circ \).