Instructor Pilot reported the nose wheel on the twin engine aircraft failed to show down and locked. The Instructor and Student Pilot performed the QRH checklist; maneuvers; and contacted company maintenance. The aircraft returned to the origin airport and landed safely.

Narrative

Shortly after getting released from departure and entering the practice area; my student and I began our training flight with slow flight. I had told my student to do the maneuver with the gear down and flaps 40. She selected the gear handle down around 130 KIAS and the nose gear presented a red circle indication. The main gear both indicated green. I had told my student to slow down to around 100 KIAS and to retract the gear and then extend it at a lower airspeed. We received the same indication and then tried the same thing but slowed down to 80 KIAS and received the same indication. I had noticed that the nose gear went from the hollow white circle straight to the filled red circle indication without going to the normal hashed box indication while it was in transit. It also took 15-20 seconds for the red indication to appear after selecting the gear handle in the down position. We referenced the checklist for gear malfunctions and determined that since our gear pump was working as intended; it was likely an issue either with the nose gear down limit switch or something else was preventing it from locking into place. On each extension; we could see the nose gear extended in the mirror on the left engine nacelle. It appeared to be normal in the mirror; it did not appear to be swinging loose. Before attempting a priority gear extension; we decided to do the maneuvers for the lesson that we could while leaving the gear up. After that; we extended the gear around 90 KIAS and got the same red nose gear indication. I thought back to something I had learned in ground school which was to try and 'yank' the aircraft around a little bit to see if we could get the nose gear to lock into place. We tried steep turns and accelerated stalls to see if that would do anything and we had no luck. Then we executed the priority gear extension checklist. We were not sure if we should have pulled the gear pump circuit breaker and then dropped all 3 from the belly of the aircraft or if we should have put the gear down and then pull the priority extension knob. We quickly deliberated and agreed to follow the checklist as it was written and then put the gear handle down; let the gear go down as is and then pull the gear extension knob. It did not fix our issue and we still had the red indication on the nose gear. We left the gear down and the gear extension knob extended as we turned back to ZZZ. After making the decision to return; I first notified the ZZZ1 around XA:55z and then called approach and informed them that we would be inbound with a gear issue and that we would need to shutdown on the runway if they could coordinate with tower. Initially we were asked if we wanted XXL or XXR and if we wanted to declare priority. We declined the priority declaration but restated we would need to shut down and be tugged off the runway and that we were fine with XXR as we were coming from the northeast in practice area Oscar. Approach then told us that tower wanted us on XXL and to proceed direct ZZZ2 Tower so we complied. ATC then declared priority for us and asked for fuel on board and souls on board. It was about a 5-7 minute flight from our location to ZZZ2 Tower so I informed my student that I would have her run the checklists for me and I was on the controls. I also wanted her to back me up to make sure I remained below 140 KIAS. We operated very well as a crew. I briefed her my plan to make the landing smooth and to use little to no steering with the rudders since the malfunction was with our nose gear. I also wanted to make sure that if she felt we were unstabilized or if she saw something that I didn't; she could call a go around. Well before ZZZ2 Tower; we both felt we were on the same page and we had briefed everything appropriately. After contacting tower; they asked if we wanted to do a low pass and we accepted the offer. I briefed to my student I would treat it as a normal landing but we would level off at 200' AGL and wait for tower tostate what position the gear appeared to be in. They indicated that the gear appeared to be down. We made another identical traffic pattern and landed smoothly with little braking and nosewheel steering. Centerline control was decent until we got to around 20 KIAS and then we slightly drifted off as we came to a stop around A3. We then shutdown on the runway; gave a quick brief to maintenance and line before they tugged the airplane off. I think it was an issue with a switch on the nose gear. As best as I could tell without speaking to maintenance; it was locked in place because we landed without it collapsing or anything like that. I had thought that I had this issue in this tail number previously where I just cycled the gear up and down and it had fixed it. But I'm not confident it was the same tail number. I'm not sure if anything could have prevented it. There may have been some technique that my student and I were unfamiliar with to get the gear to indicate green. Perhaps we incorrectly executed the priority gear extension checklist and could have gotten 3 green if we were supposed to pull the priority gear extension knob with all 3 gear up. Otherwise if the issue was with the downlimit switch; I'm not sure what could have prevented it other than having a functional downlimit switch.