The Abeian Chern-Simons theory you write down, can be written in a very generic form with a symmetric bilinear integer matrix $K_{IJ}$ with a path integral (or partition function):
$$
Z=\int DA \exp[i S]=\int DA \exp[i \int_{M^3} \frac{K_{IJ}}{4\pi} A_I dA_J]
$$
The $\int DA$ integrates all possible $A$ configuration, while $\int_{M^3}$ integrates over the spacetime manifold. Please make sure that the level quantization $K_{IJ}$ and $4 \pi$ are in the correct form.

The ground state degeneracy ($\equiv Gsd$) on a spatial manifold $M^d$ and a time $S^1$ can be computed from the partition function as
$$
Z(M^d \times S^1)=\dim(\text{Hilbert space on } M^d)\equiv Gsd_{M^d}.
$$
Below I give examples for various symmetric bilinear integer matrix $K_{IJ}$, and provide their ground state degeneracy on $T^2$ ($\equiv Gsd_{T^2}$),

$$
Z(T^2 \times S^1)=\dim(\text{Hilbert space on } T^2)\equiv Gsd_{T^2}.
$$

- For $S=\int\frac{k}{4\pi} A dA$, with $k$ is an odd integer, it is a
**fermionic** Chern-Simons TQFT which requires the manifold with a spin structure (i.e. spin manifold) to define the theory. It is **fermionic** in the sense that this TQFT can be realized in a Fermionic many-body quantum system (such as electrons, like filling-fraction 1/3 Langhlin quantum Hall state).

$$
Gsd_{T^2}=k.
$$

- For $S=\int\frac{2k}{4\pi} A dA=\int\frac{k}{2\pi} A dA$, with $2k$ is an even integer, it is a
**bosonic** Chern-Simons TQFT where the manifold with or without a spin structure can both define the theory. It is **bosonic** in the sense that this TQFT can be realized in a Bosonic many-body quantum system. For example the $Z_2$ topological order of Wen and $Z_2$ toric code of Kitaev, they are basically the same (bosonic) $Z_2$ gauge theory.

$$
Gsd_{T^2}=2k.
$$

- For the most generic $S=\int\frac{K_{IJ}}{4\pi} A_I dA_J$, their ground state degeneracy on $T^2$ ($\equiv Gsd_{T^2}$),
$$
Gsd_{T^2}=|\det(K)|.
$$
is determined by the determinant of $K$.

(1) For a more mathematical physics calculation, you may look for Belov-Moore: Classification of abelian spin Chern-Simons theories and Refs therein (those of Witten's). I believe that they provide the mathematics/differential geometry method and cite the Refs.

(2) For physics, you can check those Refs involving counting the ground states via line operators or via the distinct classes of quantum wavefunction in the Hilbert space: such as Classification of Abelian quantum Hall states and matrix formulation of topological fluids: PhysRevB.46.2290,
and Degeneracy of Topological Order: PhysRevB.91.125124

See also the more involved various non-Abelian Cherm-Simons theory Gsd and dimensions of Hilbert space.