Given that triangles \( \triangle BCE \sim \triangle JKL \), corresponding angles in similar triangles are equal.
In triangle \( BCE \):
- \( m\angle B = 89^\circ \)
- \( m\angle C = 13^\circ \)
- \( m\angle E = 78^\circ \)
For triangle \( JKL \):
- The angles correspond to the angles in triangle \( BCE \):
- \( m\angle J \) corresponds to \( m\angle B \)
- \( m\angle K \) corresponds to \( m\angle C \)
- \( m\angle L \) corresponds to \( m\angle E \)
Thus:
- \( m\angle J = m\angle B = 89^\circ \)
- \( m\angle K = m\angle C = 13^\circ \)
- \( m\angle L = m\angle E = 78^\circ \)
So the measure of angle \( J \) is: \[ m\angle J = 89^\circ \]