Light emitting diodes include a diode region having first and second opposing faces that include therein an n-type layer and a p-type layer, an anode contact that ohmically contacts the p-type layer and extends on the first face, and a cathode contact that ohmically contacts the n-type layer and also extends on the first face. The anode contact and/or the cathode contact may further provide a hybrid reflective structure on the first face that is configured to reflect substantially all light that emerges from the first face back into the first face. Related fabrication methods are also described.