I have read letters and journals of American servicemen dated to 1945 where they were already using the word "Kamikaze". What I want to know is where the American s first heard that term and how they came to use it.
Historically speaking, kamikaze (神風, meaning "divine wind") primarily referred to the storms which, on two separate occasions (in 1274 and 1281, respectively) destroyed the Mongol fleets planning on invading Japan. The term had been cemented in Japanese culture and history since then.
The official name of the air units used for suicide attacks against allied ships was "神風特別攻撃隊", meaning "divine wind special attack units". However, the pronunciation of the kanji "神風" in this name was not officially kamikaze (the kunyomi, or traditional Japanese reading), but was instead pronounced shinpū, which is the onyomi (the Japanese approximation of the original Chinese reading of the kanji).
In spite of this being the case, because the characters are the same and the meaning is shared, Japanese media and news outlets would often informally use the pronunciation kamikaze to describe these units and their method of attacking. As well as this, Americans translating Japanese news articles into English most likely will have used the kunyomi reading, due to it being the more traditional reading. Any sailor or soldier (or American citizen, for that matter) who was reading about the war in the Pacific will have been made familiar with the term kamikaze due to this.
Edit: Here's a newsreel by British Pathé from 1945 which makes use of the term. And I've also found a newspaper clipping about the attack on the USS Ticonderoga which refers to them as "Kamikaze Corps".
Double Edit: Thank you for le epic silver, kind stranger!