Since triangles \( \triangle PQR \) and \( \triangle LMN \) are similar (\( \triangle PQR \sim \triangle LMN \)), their corresponding angles are equal.
We first need to find the measure of angle \( P \) in triangle \( PQR \):
The sum of the angles in any triangle is \( 180^\circ \). Therefore, we can calculate \( m\angle P \):
\[ m\angle P + m\angle Q + m\angle R = 180^\circ \]
Substituting the given values for \( m\angle Q \) and \( m\angle R \):
\[ m\angle P + 113^\circ + 26^\circ = 180^\circ \]
Combine the known angles:
\[ m\angle P + 139^\circ = 180^\circ \]
Now, isolate \( m\angle P \):
\[ m\angle P = 180^\circ - 139^\circ = 41^\circ \]
Since \( \triangle PQR \sim \triangle LMN \), we have:
\[ m\angle L = m\angle P = 41^\circ \]
Now we can find \( m\angle N \) in triangle \( LMN \):
\[ m\angle L + m\angle M + m\angle N = 180^\circ \]
The corresponding angle \( m\angle M \) corresponds to \( m\angle Q \), which is \( 113^\circ \):
\[ 41^\circ + 113^\circ + m\angle N = 180^\circ \]
Combine the known angles:
\[ 154^\circ + m\angle N = 180^\circ \]
Now isolate \( m\angle N \):
\[ m\angle N = 180^\circ - 154^\circ = 26^\circ \]
So, the measure of angle \( N \) is:
\[ \boxed{26^\circ} \]