According to the pilot, the airplane arrived at the cross country airport late in the day, but still under daylight conditions with the runway lights illuminated. Winds were calm. The pilot maintained his target approach airspeed at 30 degrees' flaps; however, once over the runway, the airplane floated about 1,000 feet. The airplane then bounced twice with the pilot maintaining the same yoke position. After the second bounce, the pilot thought about adding power, but events were occurring too rapidly. Instead, the airplane descended again to the runway; the nose landing gear separated from the airframe, damaging the engine firewall; and the airplane slid to a stop. Neither the pilot nor the passenger reported any preexisting mechanical anomalies.