The immortal kite design!

Geometry Level 4

In the above quadrilateral \(ABCD\), \(AB=3\text{ m}\), \(BC=4\text{ m}\), \(CD=12\text{ m}\), \(AD=13\text{ m}\) and angle \(ABC=90^{\circ}\). If the angle \(BAD\) is expressible in the form \(\tan^{-1}(-\frac{a}{b})\), where \(a\) and \(b\) are coprime positive integers, find \(b-a\).

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