Since triangles \( \triangle BCE \) and \( \triangle JKL \) are similar (denoted as \( \triangle BCE \sim \triangle JKL \)), the corresponding angles in the two triangles are equal.
Given:
- \( m\angle B = 89^\circ \)
- \( m\angle C = 13^\circ \)
- \( m\angle E = 78^\circ \)
In \(\triangle BCE\): \[ m\angle B + m\angle C + m\angle E = 89^\circ + 13^\circ + 78^\circ = 180^\circ \]
To find \( m\angle J \), we note that it corresponds to \( m\angle B \): \[ m\angle J = m\angle B = 89^\circ \]
Thus, the value of \( m\angle J \) is: \[ \boxed{89} \] degrees.