In a recent paper (Groenewald et al. 2021 9) we considered an unbounded Toeplitz-like operator TΩ generated by a rational matrix function Ω that has poles on the unit circle T of the complex plane. A Wiener-Hopf type factorization was proved and this factorization was used to determine some Fredholm properties of the operator TΩ, including the Fredholm index. Due to the lower triangular structure (rather than diagonal) of the middle term in the Wiener-Hopf type factorization and the lack of uniqueness, it is not straightforward to determine the dimension of the kernel of TΩ from this factorization, and hence of the co-kernel, even when TΩ is Fredholm. In the current paper we provide a formula for the dimension of the kernel of TΩ under an additional assumption on the Wiener-Hopf type factorization. In the case that Ω is a 2×2 matrix function, a characterization of the kernel of the middle factor of the Wiener-Hopf type factorization is given and in many cases a formula for the dimension of the kernel is obtained. The characterization of the kernel of the middle factor for the 2×2 case is partially extended to the case of matrix functions of arbitrary size.