PA28 Flight Instructor reported a hard landing resulting in the propeller hitting the ground. Flight taxied back to the ramp.

Narrative

Me and my student took off during our flight slot after completing a full preflight and checking current weather conditions. On the ground prior to the flight; winds were around 11 knots however once we got in the air ATC alerted us that they were now 14 gusting to 19 knots. My student hasn't had much practice with landings and had been struggling with the crosswinds especially after we switched runways at ATC's request. We had one approach that was very unstable so I had my student go around and was holding my hands close to the controls ready to take over. Towards the end of the flight he was keeping the approach very stable coming into land but made poor judgement of our height above the ground and flared early. I also wasn't holding my hands close enough to take over in time to go around. Airspeed was already slow so we dropped and I added power to go around. However the power was added too late and we slammed into the ground which set the ELT off and resulted in a prop strike. Neither me or my student noticed the ELT going off or the bent prop. When we contacted ground control taxing back to the ramp we recognized the ELT going off. We heard the beeping when we changed frequencies and ATC let us know our ELT was going off so we reset shortly after. After parking we completed a postflight walkaround of the aircraft and secured the aircraft (chocks); especially paying attention to the wheels for any bald spots but we failed to notice the bent propeller during the walkaround.