In the previous paper hep-th/0604112 we calculated the first of the five planar two-loop diagrams for the Lcc vertex of the general non-Abelian Yang-Mills theory, the vertex which allows us in principle to obtain all other vertices via the Slavnov-Taylor identity. The integrand of this first diagram has a simple Lorentz structure. In this letter we present the result for the second diagram, whose integrand has a complicated Lorentz structure. The calculation is performed in the D-dimensional Euclidean position space. We initially perform one of the two integrations in the position space and then reduce the Lorentz structure to D-dimensional scalar single integrals. Some of the latter are then calculated by the uniqueness method, others by the Gegenbauer polynomial technique. The result is independent of the ultraviolet and the infrared scale. It is expressed in terms of the squares of spacetime intervals between points of the effective fields in the position space - it includes simple powers of these intervals, as well as logarithms and polylogarithms thereof, with some of the latter appearing within the Davydychev integral J(1, 1, 1). Concerning the rest of diagrams, we present the result for the contributions correponding to third, fourth and fifth diagrams without giving the details of calculation. The full result for the Lcc correlator of the effective action at the planar two-loop level is written explicitly for maximally supersymmetric Yang-Mills theory.