Since triangles \( \triangle BCE \) and \( \triangle JKL \) are similar (denoted by the symbol \( \sim \)), their corresponding angles are equal.
Given the angles in \( \triangle BCE \):
- \( m\angle B = 89^\circ \)
- \( m\angle C = 13^\circ \)
- \( m\angle E = 78^\circ \)
We can find \( m\angle J \) since it corresponds to \( m\angle B \) in triangle \( JKL \).
From the given information:
- \( m\angle J = m\angle B = 89^\circ \)
Therefore, \[ m\angle J = 89^\circ. \]