We report on experiments where solid-density Mg plasmas are created by heating with the focused output of the Linac Coherent Light Source x-ray free-electron laser. We study the K-shell emission from the helium- and lithium-like ions using Bragg crystal spectroscopy. Observation of the dielectronic satellites in lithium-like ions confirms that the M-shell electrons appear bound for these high charge states. An analysis of the intensity of these satellites indicates that when modeled with an atomic-kinetics code, the ionization potential depression model employed needs to produce depressions for these ions which lie between those predicted by the well known Stewart-Pyatt and Ecker-Kroll models. These results are largely consistent with recent density functional theory calculations.