The gallium nitride haeckelite (4|8-GaN) phase is an attractive material for a two-dimensional (2D) light-emitting diode (LED); however, its p-type doping is still challenging due to hole carrier trapping. Our density functional theory calculations suggest strain engineering as a route to release trapped hole carriers. We show that Mg and Be impurities in 4|8-GaN have multifarious hole states, including symmetry-broken polaronic (trapped) and delocalized (extended) states, whose detrapping energies are estimated to be 33.4 and 263.3 meV for Mg and Be impurities, respectively. The hole states trapped by a Mg impurity can be, however, detrapped by applying a moderate tensile strain around 2% perpendicular to the 4|8 plane, which would critically enhance p-type dopability. We further show that the photoluminescence (PL) spectrum of a Mg impurity can be tuned by the lattice strain, which enables efficient control for light emission of 4|8-GaN. Our findings pave the way to design an atomically thin blue LED based on 4|8-GaN.