1. **State the problem:** We need to find the measure of angle $\angle KML$ in quadrilateral $KJML$ where $\angle J = 108^\circ$ and $\angle MNL = 25^\circ$. 2. **Understand the setup:** The diagonals $KN$ and $ML$ intersect at point $N$. We are given $\angle MNL = 25^\circ$, which is an angle formed at the intersection of the diagonals. 3. **Key property:** When two lines intersect, opposite (vertical) angles are equal. Since $\angle MNL = 25^\circ$, the vertical angle opposite to it at $N$ is also $25^\circ$. 4. **Use the quadrilateral angle sum:** The sum of interior angles in any quadrilateral is $360^\circ$. 5. **Calculate $\angle KML$:** Given $\angle J = 108^\circ$, and knowing the relationship between the angles at $N$ and $M$, we can deduce $\angle KML$. 6. **Since $\angle MNL$ and $\angle KML$ are vertical angles, they are equal:** $$\angle KML = \angle MNL = 25^\circ$$ **Final answer:** $$m\angle KML = 25^\circ$$