  We study indirect exchange interactions between localized spins of magnetic
impurities in spin-valley coupled systems described with the Kane-Mele model.
Our model captures the main ingredients of the energy bands of 1H transition
metal dichalcogenides (TMDs) monolayers, such as 1H-MoS$_2$ and 1H-NbSe$_2$. To
obtain the effective interactions, we use the exact diagonalization of the
Hamiltonian, avoiding momentum cut-offs. We start by comparing the standard
perturbation expansion in terms of the Kondo exchange with the exact
calculation of the interaction, treating the local spins classically. We find
that perturbation theory works well even beyond the regime where the relevant
figure of merit, the ratio between the exchange $J$ and the hopping $t$, is
small. We verify that the effective indirect exchange Hamiltonian derived from
perturbation theory also works in the non-perturbative regime. Additionally, we
analyse the interplay between the symmetry of the different terms of the
interaction (Heisenberg, Ising, and Dzyaloshinskii$-$Moriya (DM)), the
Fermi-surface topology, and the crystallographic direction in which the
impurities are placed. We show that the indirect exchange along the armchair
direction is actually Heisenberg-like, due to the reflection symmetry of the
crystal structure around this direction. Finally, we explore the exploitation
of indirect exchange, combined with atomic manipulation, to engineer the
Majumdar-Ghosh Model. Our results show that TMDs provide an extremely versatile
platform to engineer indirect exchange interactions.
