PA-24 pilot reported having trouble landing and multiple go-around attempts due to shuddering. On the third attempt and upon finally landing; the pilot shut down the aircraft and saw that the wheel hub of the nose gear had broke and a prop strike had occurred.

Narrative

I flew Aircraft X from ZZZ to ZZZ1 direct. On takeoff; I encountered an issue where the passenger side door popped open in flight. We diverted to ZZZ2 and made an uneventful landing on Runway XX. The door was closed during the stop-and-go. The remainder of the flight was uneventful. Upon making visual approach to ZZZ1 Runway XX; we attempted a landing. The main wheels touched down at about 90 mph; and when I let the nose drop; I felt an intense shuddering that I have never felt before. I then pulled back attempting to relieve stress on the nose wheel; which caused the plane to become airborne again. Worried about a belly flop; I executed a go-around. ZZZ1 Tower asked me to begin left traffic; and I then began another landing attempt. I decided to keep a more nose-up attitude during the flare to relieve stress on the nose gear. Unfortunately; this caused me to bounce on the main wheels and I was not able to get the plane on the ground. I executed a second go-around. Suspecting that the shuddering had something to do with the landing struts; I asked the Tower Controller if they could take a look at my landing struts. The controller replied that they did not know what struts were; so I focused on running the landing checklist again. On the third landing attempt; I came in over the numbers nearly at stall speed; at full flaps; deciding that if I didn't trust the landing gear I just wanted to be as slow as possible. I don't remember what happened during this landing. We definitely bounced multiple times; but the plane eventually landed quite short and we taxied off the runway under our own power. I told ATC we have an issue with the landing gear due to the shuddering; and so I taxied to the ramp and shut off. After getting off; I saw that the wheel hub of the nose gear had disintegrated and that there had been a prop strike.