Since triangles \( \triangle BCE \) and \( \triangle JKL \) are similar (denoted as \( \triangle BCE \sim \triangle JKL \)), it follows that corresponding angles are equal.
Given:
- \( m\angle B = 89^\circ \)
- \( m\angle C = 13^\circ \)
- \( m\angle E = 78^\circ \)
From the properties of triangles, we know that the sum of angles in a triangle is \( 180^\circ \): \[ m\angle B + m\angle C + m\angle E = 180^\circ \]
Let's confirm this: \[ 89^\circ + 13^\circ + 78^\circ = 180^\circ \] This is correct.
Now, in similar triangles, \( \angle B \) corresponds to \( \angle J \). Therefore, we have: \[ m\angle J = m\angle B = 89^\circ \]
Thus, the measure of angle \( J \) is: \[ \boxed{89^\circ} \]