This follows from Remark 4.17 / Proposition 4.10 of "Cuts and flows of cell complexes" by Duval, Klivans and Martin. Their formula (as applied to your problem, by taking $\Sigma=\Upsilon=K$) states:

$$
\det(\partial^t_2\partial_2)=\sum_\Gamma |\mathbf{T}(\tilde H_{1}(K,\Gamma))|^2.
$$

Here $\mathbf{T}(\cdot)$ denotes the torsion subgroup and the sum over $\Gamma$ runs over all "relatively acyclic" subcomplexes of $K$ (Definition 3.1). Since $\tilde{H}_1(K)$ is torsion, the discussion before Proposition 3.2 shows that the $\Gamma$ are in fact the spanning trees of $G$.

Furthermore, the relative homology groups $\tilde{H}_1(K,\Gamma)$ are all isomorphic to $\tilde{H}_1(K)=\mathbb{Z}_m$ because all the $\Gamma$'s are trees and hence contractible. The result is that the sum evaluates to $m^2 k(G)$ as desired.

Here follow a few general comments. The above formula is an interesting higher-dimensional variant of the Kirchhoff Matrix-Tree theorem which expresses the number of spanning trees of a graph in terms of the determinant of the "up-down" Laplacian $L^{ud}=\partial\partial^t$ (with a row and vertex removed). Contrast this with the "down-up Laplacian" $L^{du}=\partial_i^t\partial_i$ that's considered here. These two matrices are closely related (physicists might call them "supersymmetric partners") since the multisets of their nonzero eigenvalues are identical. However, it seems that $L^{ud}$ shows up more frequently in practice and hence is much more studied. I could be showing my ignorance but other than the above formula, I haven't seen much else on $L^{du}$ in the literature.

The "up-down / down-up" terminology can be found in Duval, Klivans and Martin's survey on simplicial and cellular trees which includes several higher-dimensional generalizations of Kirchhoff (all involving $L^{ud}$ though!).