Flight instructor reported students unstable approach led to a hard landing that resulted in the nose gear to collapse.

Narrative

This was a normal training flight with block time of XA00-XC00 in Aircraft X. Preflight briefing was conducted and consisted of maneuvers and emergency procedures. I told the student that I wanted to see power-on stalls; power off stalls; slow flight; steep turns and Engine Failure emergencies. If time permitted; I wanted to see some short field take offs and landings. Takeoff and landings data was discussed and numbers were correct. The assessment form (PAVE and IMSAFE) was filled out and signed. His score was within normal parameters. Preflight and engine start was normal. We taxied out to runway 33 and engine run up was normal according to the POH. We conducted a normal take off with no problems noted. WE left the traffic pattern on the upwind departure 45 degree angel and departed over to the practice area. We worked on briefed maneuvers until they were within ACS (Airman Certification) standards for the Sport Pilot Rating and departed to the ZZZ [airport] at approx 1500ft. First pattern at ZZZ resulted in a go-around. The reasons were 1) started base leg too soon; and too high; 2) improper configurations at the base turn; 3) airspeed was too high turning to final and 4) the student did not identify a forward slip was needed. On the next pattern; he had proper downwind spacing; but stabilized 200ft below traffic pattern altitude and remained there. He properly configured the aircraft prior to the base turn and turned base late resulting in a ling final and low on roll out. He did not realize he was low and I had to tell him to correct his pattern. The student leveled off; and anticipated properly coming back to the glide and path and maintained proper speed. He was fully configured for landing approx. 500 AGL.ZZZ is an airport with an elevation of 80ft; with one runway XX and XY. It is 4;000ft long by 50ft wide. When the student crossed the threshold he was on the appropriate approach speed but was too high. He proceeded to flare high and reduced the power to idle; which caused him to float. To correct the floating; he pushed the nose over and created a high sink rate. He continued on to touchdown and landed firmly; and then bounced on all three wheels. The student did not execute the go around; so I took controls; executed the go around and flew the aircraft back to Orlando Apopka Airport (X04). At X04; I entered the downwind and flew a normal pattern. Touchdown was normal; roll out was normal with no nose wheel vibration noted. I wasted the runway on to the taxiway at approximately 7 KTS of ground speed. Halway through the turn off and clear of the runway; the nose wheel collapsed. I immediately shut down the aircraft and made a radio call to incoming traffic prior to evacuating the aircraft.