Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$. Find $\angle KAL$.
1. Consider the square \(ABCD\) with points \(K\) and \(L\) on sides \(BC\) and \(CD\) respectively.
2. Given that \(\angle AKB = \angle AKL\), we need to find \(\angle KAL\).
3. Since \(ABCD\) is a square, all its internal angles are \(90^\circ\).
4. Let \( \angle AKB = \angle AKL = \theta \).
5. Note that \(\angle AKB\) and \(\angle AKL\) are angles formed by the lines \(AK\) with sides \(BC\) and \(CD\) respectively.
6. Since \(A\) is the excentre of \(\triangle KLC\), we can use the properties of the excentre and the angles in a square.
7. In a square, the diagonals are equal and bisect each other at \(90^\circ\). Therefore, the angle between the diagonal and any side is \(45^\circ\).
8. Since \(\angle AKB = \angle AKL\), and considering the symmetry and properties of the square, we can deduce that \(\angle KAL\) must be half of the right angle formed by the diagonals.
9. Therefore, \(\angle KAL = 45^\circ\).

The final answer is \(\boxed{45^\circ}\).