Let \(H^2\) be the Hardy space over the bidisk with variables \(z\) and \(w\). Denote by \(T_z\) and \(T_w\) the multiplication operators on \(H^2\) induced by \(z,w\), respectively. A nontrivial proper closed subspace \(N\subset H^2\) is said to be a mixed invariant subspace if \(T_zN\subseteq N\) and \(T_{w}^* N \subseteq N\). Let \(V_z\) and \(V_w\) be the compressions of \(T_z\) and \(T_w\), respectively, to \(N\). In this paper, the authors study mixed invariant subspaces for which the rank of the commutator \([V_z,V_w]\) is one and the dimension of \(N\ominus zN\) is two (mixed invariant subspaces \(N\) for which \(\operatorname{rank}[V_z,V_w]\leq 1\) and \(\dim(N\ominus zN)=1\) have been studied in Part I [K. J. Izuchi and M. Naito, Nihonkai Math. J. 20, No. 2, 145–154 (2009; Zbl 1200.47010)]). Since it is assumed that \(\dim(N\ominus zN)=2\), one can write \(N\ominus zN={\mathbb C}\,F\oplus {\mathbb C}\,G\), where \(F, G\in H^2\). The main result of the paper is the description of these functions.