The airplane experienced a landing gear collapse while on the landing roll. The certified flight instructor (CFI) reported that she was performing a landing in the multiengine airplane. The student kept his hands on the control yoke in an attempt to follow along with the instructor and feel her control inputs. While approaching the runway surface, the CFI configured the airplane for a power-off landing and retarded the throttles aft. She noted that all three landing gear lights in the cockpit were illuminated green in indication that landing gear was in a down and locked position. She confirmed this configuration visually by looking at the reflection of the extended landing gear in a small mirror located on the wing. Upon touchdown, the main landing gear made contact with the runway and the student pilot began to reach for the landing gear handle thinking that it was the flap handle. The CFI waved her hand to block him and told him "no." The nose landing gear collapsed and the airplane slid on the nose. After the airplane came to rest the CFI noted that the landing gear handle was in the "down" position. The student pilot stated that he did not touch the landing gear handle. A mechanic repairing the airplane reported that the landing gear and linkages were all intact with no anomalies noted. The actuator was not tested due to the extent of the damage. The airplane incurred damage to a rib in the nose.