Student Pilot reported propeller damage after a hard landing on a solo flight.

Narrative

I was on my 3-point solo cross-country flight as a student pilot. I departed from ZZZ1 and my plan was to land at ZZZ2 as well as ZZZ before returning to ZZZ1. Roughly 20 nm southwest inbound for ZZZ2; I attempted to listen to the ATIS and I was not getting anything. I assured myself I was on the right frequencies using my notes; and flight plan; as well as double-checking with foreflight concluding the comm 2 was just not working. The radio was fully functioning as I retrieved the weather via the comm 2 in the aircraft before I departed ZZZ1. Approximately 12 nm out from the airport; I contacted ZZZ Tower and notified them I was inbound with no weather because of radio issues. The Tower told me the current winds and plan of action for landing which I executed but given that I did not have the same amount of time as I was familiar with to evaluate the current winds to the best of my ability ended up being a factor that led up to the propeller strike. I reported midfield on left downwind at 2;000 ft. as instructed by the Tower and began my descent at my discretion. The winds were extremely variable as I descended below pattern altitude and as I lined up on base and eventually final I found myself fighting against the uneasy wind and poor aircraft performance. As I touched down it felt as if I had landed normally but I began to balloon where the plane wanted to take off and the aircraft uncontrollably was bouncing. The nose wheel gave out causing me to completely lose control of the aircraft leading to the propeller strike. Looking back at it I could have performed a go-around. Initially directed by the Tower to either do a touch and go or stop and go on the Runway I knew the airport was busy and I felt it was necessary to execute the first landing to avoid other traffic backups as I did end up with clearance to perform a full stop taxi back.