Mg-doped InGaN/GaN p-type short-period superlattices (SPSLs) are developed for hole injection and contact layers of green light-emitting diodes (LEDs). V-defect-related pits, which are commonly found in an InGaN bulk layer, can be eliminated in an InGaN/GaN superlattice with thickness and average composition comparable to those of the bulk InGaN layer. Mg-doped InGaN/GaN SPSLs show significantly improved electrical properties with resistivity as low as $0.35 ohm-cm, which is lower than that of GaN:Mg and InGaN:Mg bulk layers grown under optimized growth conditions. Green LEDs employing Mg-doped InGaN/GaN SPSLs for hole injection and contact layers have significantly lower reverse leakage current, which is considered to be attributed to improved surface morphology. The peak electroluminescence intensity of LEDs with a SPSL is compared to that with InGaN:Mg bulk hole injection and contact layers.