Artificially frustrated spin-ice systems are of considerable interest since they simulate the spin frustration and concomitant rich behavior exhibited by atoms on a crystal lattice in naturally occurring spin-ice systems such as pyrochlores. As a result of the magnetic frustration, these systems can exhibit 'magnetic monopole' type defects, which are an example of an exotic emergent quasiparticle. The local magnetization structure of such monopole defects determines their stability and thus is critical to understanding their behavior. In this paper, we report on the direct observation at room temperature of the nanoscale magnetic structure of individual magnetic monopoles in an artificially frustrated two-dimensional square spin-ice lattice, using high-resolution aberration-corrected Lorentz transmission electron microscopy. By combining the high-resolution microscopy with micromagnetic simulation, we demonstrate how nucleation of defect strings, reminiscent of Dirac strings, connecting monopole defects controls the demagnetization process in these spin-ice lattices.