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The kagome lattice can host exotic magnetic phases arising from frustrated and competing magnetic interactions. However, relatively few insulating kagome materials exhibit incommensurate magnetic ordering. Here, we present a study of the magnetic structures and interactions of antiferromagnetic Na$_2$Mn$_3$Cl$_8$ with an undistorted Mn$^{2+}$ kagome network. Using neutron-diffraction and bulk magnetic measurements, we show that Na$_2$Mn$_3$Cl$_8$ hosts two different incommensurate magnetic states, which develop at $T_{N1} = 1.6$ K and $T_{N2} = 0.6$ K. Magnetic Rietveld refinements indicate magnetic propagation vectors of the form $\mathbf{q} = (q_{x},q_{y},\frac{3}{2})$, and our neutron-diffraction data can be well described by cycloidal magnetic structures. By optimizing exchange parameters against magnetic diffuse-scattering data, we show that the spin Hamiltonian contains ferromagnetic nearest-neighbor and antiferromagnetic third-neighbor Heisenberg interactions, with a significant contribution from long-ranged dipolar coupling. This experimentally-determined interaction model is compared with density-functional-theory simulations. Using classical Monte Carlo simulations, we show that these competing interactions explain the experimental observation of multiple incommensurate magnetic phases and may stabilize multi-$\mathbf{q}$ states. Our results expand the known range of magnetic behavior on the kagome lattice., Comment: 13 pages, 8 figures