Suli's geometry problem

Geometry Level 5

Let \(ABC\) be a triangle with \(AB = 17, AC = 25, BC = 28 \). The triangle has circumcircle \(\Gamma\) and incenter \(I\). Point \(D\) is on \(\Gamma\) such that \(\angle ADI = 90^{\circ}\), and extend \(AI\) to meet \(BC\) at \(E\). If \(\sin^2 \angle IDE\) can be represented as \(\frac{m}{n}\) with \(m\) and \(n\) are coprime positive integers, compute \(m+n\).

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