Using KCL/KVL to derive differential equation of a mixed RLC-ciruit
The thing you got wrong here is the Kirchhoff's laws : while you're right on the current laws, you made a little mistake on the voltages : they are equivalent in each parallel wire, so that :
$$
v(t) = v_{C1}(t) = v_{L1}(t) = v_R(t)+v_{C2}(t)+v_{L2}(t)
$$
Note that these equations are given without regard to the sign and the conventions.
This way should lead you to the proper solution.
Always remember : the Kirchhoff's laws says that the sum of all the voltages in a loop (closed network) is equal to zero, and that the sum of all currents in a node are equal to zero.