First step is to write down the given information in terms of SI units, so you have:
Next step is to write down all the forces, i will pick the left sphere, although either is fine.
so the x component of the force is: Fx= T*sin(theta)-Fe and hence,
T*sin(theta) = Fe(where T is the tension in the string, and Fe is the electric force)
Now write down y component of forces: Fy= T*cos(theta)-mg = 0 and hence, T*cos(theta) = mg (where m is mass and g is gravity=9.8 m/s)
Now remember your goal is to solve for L, to do so would require knowing the length between the two charges, or r.
To find r, we must first solve for Fe, which we can do by the following:
take Fx and divide by Fy, resulting in: tan(theta) = Fe/mg (notice the T cancels out)
So solving for Fe results in, Fe = mg*tan(theta)
plug in your values and you get 1.65*10^-4 N as Fe.
Now you can solve for r, by plugging this value into the electric force formula which is:
F=(ke * q1*q2)/ r^2
where ke=8.988*10^9 N*m^2/C^2
your only unknown is r, so rearrange the equation until you get r by itself:
r = squareroot of ((ke*q1q2)/Fe)
and so upon substituting in your values you get .05757m for r. However recall that this is actually the distance from the 2 charges, and what we are looking for is half of that, so lets call that value a, and a = .028785 m, which is the distance from one charge to the middle.
Now to solve for L, we look at the geometry of the triangle and come to the conclusion that sin(theta) = opposite / hypothenuse, and hence
sin(theta) = a/L
so L = a/sin(theta)
Plug in the values and solve.