We are given that:

• $JL = x + 10$
• $KL = 4x$
• $JK = 2x$

Since $K$ is a point on the line segment $JL$, we can use the fact that the sum of the lengths of the segments $JK$ and $KL$ equals the total length of $JL$. Therefore, we can set up the equation:

$JK + KL = JL$

Substituting the given expressions:

$2x + 4x = x + 10$

Simplifying the left-hand side:

$6x = x + 10$

Now, subtract $x$ from both sides:

$5x = 10$

Solving for $x$:

$x = 2$

Now that we have $x = 2$, we can find the numerical length of $KL$:

$KL = 4x = 4(2) = 8$

Thus, the length of $KL$ is $\boxed{8}$.

Would you like further details or have any questions? Here are five related questions:

1. What is the length of the entire segment $JL$ when $x = 2$?
2. What is the length of segment $JK$ when $x = 2$?
3. How do you verify that the total length of $JL$ is consistent with the sum of $JK$ and $KL$?
4. If point $K$ was shifted, how would it affect the lengths of $JK$ and $KL$?
5. Can we generalize the method for any point $K$ on a line segment $JL$?

Tip: Always check your solution by substituting the values back into the original equation to ensure consistency.