Design a two-stage LC low-pass filter with one fixed stage
I have derived the transfer function of this 4th-order network but the trick is to factor it into two second-order polynomial forms. As there are no zeros in this network, we can use the Fast Analytical Techniques (FACTs) to determine the denominator coefficients from \$b_1\$ to \$b_4\$ or use a simple Thévenin approach and ask Mathcad to factor the \$s\$ terms. The only difference is that Mathcad won't arrange terms in series-parallel forms what the FACTs would let you do. Anyway, here, the transfer function came quite easily. Below is the Mathcad sheet showing the raw transfer function (using Thévenin) and the other one using a 4th-order polynomial form:



The difficulty now lies in trying to factor-in the 4th-order denominator into two 2nd-order filters. If we consider that the second resonance is above the first one, then a simplification is feasible. This is what the below sheet shows and the deviation with the full-blown formula is not too ugly. Considering your fixed values for \$L_6\$ and \$C_4\$, you should be able to extract the values for \$L_8\$ and \$C_5\$.



As usual, I encourage you to discover the FACTs which are of invaluable help for determining transfer functions for passive or active circuits. You can start with the Extra-Element Theorem from Dr. Middlebrook (https://en.wikipedia.org/wiki/Extra_element_theorem) and look at the presentation taught at APEC in 2016: http://cbasso.pagesperso-orange.fr/Downloads/PPTs/Chris%20Basso%20APEC%20seminar%202016.pdf

You also have an example of a 4th-order \$LC\$ network tuned for a maximally-flat response here http://cbasso.pagesperso-orange.fr/Downloads/Book/List%20of%20FACTs%20examples.pdf.