Hole in the cylinder

First, Let us consider a cylinder with NO CAVITY

Using Gauss Law, It can be shown that the electric field at a distance $2L$ is $E_1= \dfrac{\rho L}{4 \epsilon_0}$ .

Similarly, For a sphere of radius $\dfrac{L}{2}$ ,the electric field at a distance $2L$ is $E_2 = \dfrac { \rho L}{96 \epsilon_0}$ .

By the principle of superposition, The Field at a distance $2L$ is the resultant field due to the cylinder minus the sphere.

That is,

$E= E_1-E_2$

On simplification,we get

$E= \boxed{\dfrac{23}{96} \dfrac{\rho L}{\epsilon_0}}$

So $a+b= \boxed{\boxed{119}}$