Flight Instructor reported their student lost control of the aircraft during landing resulting in a bent propeller and nose gear damage.

Narrative

During training; the final approach was stabilized. Upon crossing the threshold; the student was instructed to level out and reduce power. Instead the student continued pitch down without any power adjustment which resulted in a nose wheel landing. After landing on the nose wheel; the student reacted with an abrupt elevator control input. The instructor tried to correct hands-on but the student's control was too strong and high bouncing occurred. As the instructor loudly stated 'My controls' the student cooperated. Instructor resolved the bouncing by adding power then subsequently reducing power to settle the airplane to the ground. The tower asked if we'd like to continue or taxi off the runway. We taxied off the runway. During taxi; instructor could feel the leaning of the aircraft and extra rudder input was required to steer the nose wheel. Engine operated smoothly and normal as usual during the incident; during taxi; and during shutdown. After parking; observation found the nose wheel in damaged condition and the propeller ends slightly bent. The propeller strike was never felt or noticed during the incident. Instructor notified the owner/operator of the flight school of the incident. Mechanic visually inspected the damage; then he towed and stored the aircraft into the hangar. Monetary damage assessment TBD.