Let the sides be $13$, $x$, and $2x$.

By the triangle inequality, the sum of the lengths of any two sides must be greater than the third side.

So, we have:

1. $13 + x > 2x$
2. $13 + 2x > x$
3. $x + 2x > 13$

Let's solve each inequality:

1. $13 + x > 2x \implies 13 > x$
2. $13 + 2x > x \implies 13 + x > 0$ (which is always true for $x > 0$)
3. $x + 2x > 13 \implies 3x > 13 \implies x > \dfrac{13}{3}$

Since $x$ is an integer, $x \geq 5$.

Also, from (1), $x < 13$.

So the possible integer values for $x$ are $5, 6, 7, 8, 9, 10, 11, 12$.

There are $8$ possible values.

Answer: D) 8