To solve for \( m \angle LKN \), we can use the properties of parallel lines and the relationships between the angles formed by a transversal.
Given that lines JL and MO are parallel and the transversal is the line connecting points K and N, we have the following angles:
- \( \angle MNK = 114^\circ \)
- Angle \( LKN \) is an alternate interior angle to angle \( MNK \).
Since angle \( MNK \) and angle \( LKN \) are alternate interior angles, and since JL and MO are parallel, we know that these angles are equal.
Therefore, we have:
\[ m \angle LKN = m \angle MNK = 114^\circ \]
Thus, the measure of angle \( LKN \) is \( 114^\circ \).