PA28 pilot trainee reported the aircraft nose gear was damaged during a landing incident following an unstabilized approach.

Narrative

My flight today was a solo training cross country flight as part of my commercial pilot training at a Part 141 flight school. I preflighted and did not find anything out of the ordinary during my checks. The weather conditions were within my minimums and relatively calm. I took off from ZZZ1 at about XA:17. The taxi and takeoff were uneventful. The flight was going well. Before getting to ZZZ I contacted Approach and requested and received flight following to ZZZ2 for a touch and go and to ZZZ3 for a full stop. Both the touch and go and the full stop landing were uneventful. I landed at ZZZ2 and got a VFR transition to ZZZ3. After getting fuel I took off again. I did encounter some light turbulence my flight to ZZZ. Once I was in ZZZ's Class C airspace they asked me to make left traffic for Runway XX. The Tower had me extend my downwind for a Cessna 172 on final approach and then cleared me to land. The Tower reported winds as calm. My approach started high but I quickly got back on glide using an idle power setting for most of the approach. On short final I remember feeling a little slow but the approach was still stable. I flared when I felt it was appropriate to. The plane's main tires contacted the runway and caused the aircraft to bounce back into the air. The landing was not ideal but I felt It could be salvaged by increasing the flare. The second landing was very flat causing the nose wheel to absorb most of the contact with the runway. The next thing I remember is having the nose on the ground; the propeller stopped; and coming to a halt on the centerline of Runway XX. I immediately went through my emergency memory items and shut off the fuel pump; cut the mixture; and after contacting the Tower to request assistance proceeded with securing the aircraft. I got out of the airplane after I saw emergency vehicles approaching. The entire nose wheel assembly was later found by airport personnel many hundreds of feet behind the aircraft.Regardless of whether or not the aircraft was structurally vulnerable I feel as though I should have gone around after the bounce. It was a long flight and maybe my judgment was off. I could have added power to cushion the landing; I think I also just should have kept in mind that I had a lot of runway left and that I could have made smaller corrections that may have changed the outcome of the event.