The pilot was practicing landing maneuvers with their flight instructor. The early part of the flight was uneventful, and they decided to work on short-field landings, all of which resulted in go-arounds. They subsequently started to practice power-off 180° landings, and on the third landing attempt, the airplane was high on the approach. The student performed a forward slip maneuver, however the airplane was traveling too fast on touchdown, and it bounced. The nose then dropped, and the nosewheel struck the runway. The flight instructor took the flight controls just as the nosewheel struck the runway again and added power to initiate a go-around. The airplane sustained substantial damage during the sequence, however the airplane performance appeared unaffected and the crew continued training unaware of the damage. It was not until after landing, when the lesson was complete, that they noticed damage to the propeller tip and forward fuselage structure. The pilot and flight instructor reported no preimpact mechanical anomalies or malfunctions that could have precluded normal operation.