$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $\angle BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, find the length of chord $AC$, expressed in inches. The answer should be in the form \(\frac{k\sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers. Please calculate the value of \(k + m + n\).

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