Since the triangles \(\triangle BCE\) and \(\triangle JKL\) are similar (denoted as \(\triangle BCE \sim \triangle JKL\)), their corresponding angles are equal.
Given the angles in triangle \(BCE\):
- \(m\angle B = 89^\circ\)
- \(m\angle C = 13^\circ\)
- \(m\angle E = 78^\circ\)
Since these are the angles in triangle BCE, we can pair them with the angles in triangle JKL:
- \(m\angle B = m\angle J\)
- \(m\angle C = m\angle K\)
- \(m\angle E = m\angle L\)
From this, we have:
- \(m\angle J = m\angle B = 89^\circ\)
- \(m\angle K = m\angle C = 13^\circ\)
- \(m\angle L = m\angle E = 78^\circ\)
Thus, the measure of angle \(J\) is: \[ \boxed{89^\circ} \]