BE55 Instructor reported unreliable nose gear indication resulted in nose gear collapse when aircraft landed.

Narrative

I was flying with my student doing multi engine maneuvers. At the end of our simulated emergency descent/ smoke in the cockpit situation; we began to retract the landing gear to end the maneuver. When we put the gear selector in the up position; we felt a snap in the floor. My student looked at the mirror that was pointed at the nose gear and told me that the gear was swinging freely. We then put the gear in the down position where we received the green light in the cockpit. My student looked at the mirror and said that the gear was still swinging freely. We then began to troubleshoot any problems that we could come up with. We didn't feel like we should keep moving the gear selector back in the up position in case we would possibly lose our mains too. We then descended to try and get phone service. I texted two mechanics at my company and also my boss. They all came to the conclusion to go into ZZZ and [request priority]. We contacted ZZZ Tower and [requested priority]. As we were approaching ZZZ; ATC asked us if we wanted to do a fly by the Tower to see if they could confirm our mains were down. We then did the fly by where they were able to confirm that our gear was in the down position. We then made right traffic for Runway XX. We made right traffic and landed with a little faster airspeed and only two notches of flaps. When our mains touched down; we tried holding the nose off as long as we could while also cutting the mixture. Our nose eventually fell and we skidded across the runway until we stopped. When we came to a stop; we both unbuckled and jumped out of the plane as fast as we could.