According to the flight instructor, the student pilot was a little high on the final approach for the short field landing, which resulted in a steeper approach. Before touch down, the student decreased the pitch attitude, and there was little to no flare. The airplane landed flat, bounced, then hit the nosewheel first as they touched down and bounced a second time. During the second bounce, both pilots heard a noise, the student felt a vibration in the rudder pedals, and initiated a go-around. The flight instructor assumed control of the airplane during climb out and confirmed there were no anomalies with the rudder system before transferring the controls back to the student. During the next landing, the flight instructor stated that the student flared a little high and the airplane ballooned before touchdown. During the landing roll, the airplane veered to the right and neither pilot was able to maintain directional control. The nose landing gear collapsed as both pilots applied the brakes. The airplane impacted a taxiway light and came to rest on the runway. The airplane sustained substantial damage to the firewall and lower fuselage at the nose landing gear support structure. Based on the pilot’s statements it is likely the nose landing gear was compromised during the first landing attempt. The flight instructor reported that there were no anomalies with the airplane before the first landing. He added that the accident could have been prevented if they had performed a go-around if a stabilized approach was not established.