  The inert 2-Higgs Doublet Model (i2HDM) is a well-motivated minimal
consistent Dark Matter (DM) model, but it is rather challenging to test at the
Large Hadron Collider (LHC) in the parameter space allowed by relic density and
DM direct detection constraints. This is especially true when considering the
latest XENON 1T data on direct DM searches which we use here to present the
best current combined limit on the i2HDM parameter space. In this analysis, we
present prospects to advance the exploitation of DM mono-jet signatures from
the i2HDM at the LHC, by emphasising that a shape analysis of the missing
transverse momentum distribution allows one to sizably improve the LHC
discovery potential. As a key element of our analysis, we explore the validity
of using an effective vertex, $ggH$, for the coupling of the Higgs boson to
gluons using a full one-loop computation. We have found sizeable differences
between the two approaches, especially in the high missing transverse momentum
region, and incorporated the respective K-factors to obtain the correct
kinematical distributions. As a result, we delineate a realistic search
strategy and present the improved current and projected LHC sensitivity to the
i2HDM parameter space.
