Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$. If the area of the square $PQRS$ is $m + n\sqrt{k}$ where $m$ and $n$ are integers and $k$ is a prime number, then determine the value of $\sqrt{\frac{m+n}{k^2}}$.

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