Let $\triangle ABC$ be a triangle in a plane such that $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point in three-dimensional space such that $\angle{BDC}=\angle{CDA}=\angle{ADB}=90^\circ$. Let $d$ be the distance from $D$ to the plane containing $\triangle ABC$. The value $d^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.

Let's think step by step and output the final answer within \boxed{}.
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