Magnetic and electric constant relation special relativity

For the first time I am trying to derive by myself the general relation between the magnetic and electric constants as shown in the following:

$$ F = frac{k_E q_1 q_2}{r^2} hspace{2mm} , hspace{2mm} F = frac{k_M I_1 I_2}{r} $$

To do this, I am considering a thought experiment where I have two lines of moving charges spaced from one another by a distance $r$ with charge densities $lambda_1$ and $lambda_2$ and moving with velocity $v$ in the $x$ direction such that the corresponding currents are $I_1$ and $I_2$. Using $I_k = lambda_k v$, I calculate the total force between the wires in the rest frame as (positive pulling the wires together):

$$ F = frac{k_M lambda_1 lambda_2 v^2}{r} – frac{2k_E lambda_1 lambda_2}{r} $$

Then I consider a frame in which the observer is moving at the same velocity as the charges in the wires so that there is no current. I then, using $lambda_k^{‘} = gamma lambda_k$, I calculate the force in the moving frame as:

$$ F’ = -frac{2k_E lambda_1 lambda_2 gamma^2}{r} $$

And since the force is perpendicular to the velocity, we should have that $F = F’$. My problem is that when I set these equal to one another, I find that I don’t get the nice relation that I have seen elsewhere ($c^2 k_M = 2k_E$). I have noticed that if I instead have:

$$F’ = -frac{2k_E lambda_1 lambda_2}{gamma^2 r} $$

Then I get the correct result. However, I don’t see why that would be the case, as I believe my charge density transformation to be correct. Can anyone help me find where I have gone wrong? Thanks in advance!