Let one of the diagonals of the [rhombus](https://artofproblemsolving.com/wiki/index.php/Rhombus) be $a$ units long, so the other diagonal is $2a$ units long.  That means the area of the rhombus is $\frac{1}{2} \cdot 2a \cdot a = a^2$, so $K = a^2$.
From the [Pythagorean Theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem), one of the side lengths of the rhombus is $\sqrt{(\frac{a}{2})^2 + a^2} = \sqrt{\frac{5a^2}{4}} = \frac{\sqrt{5K}}{2}$, so the answer is $\boxed{\textbf{(E)}}$.
A rhombus is given with one diagonal twice the length of the other diagonal. 
Express the side of the rhombus is terms of $K$, where $K$ is the area of the rhombus in square inches.
$\textbf{(A)}\ \sqrt{K}\qquad \textbf{(B)}\ \frac{1}{2}\sqrt{2K}\qquad \textbf{(C)}\ \frac{1}{3}\sqrt{3K}\qquad \textbf{(D)}\ \frac{1}{4}\sqrt{4K}\qquad \textbf{(E)}\ \text{None of these are correct}$