Since "youthful immortal" is not a common reading for the name "Midori," if you have the opportunity to ask your parents their intended kanji, that would be the easiest route.
The main kanji for Midori is 緑 (meaning greenery, or the color green). There are a few obsolete kanji variations on it that hold the same meaning. Another possible combination that my Japanese grandmother gave is 美＋鳥＝美鳥 (beautiful bird).
I am a Japanese American with Midori as my middle name. My Japanese grandmother's name was Mitsu but she used Midori as her nickname/stage name. She gave my mother Midori for a middle name, and then my mother gave it to me. My grandmother immigrated to America in an era where most women's names were written in katakana with no kanji (most of the women in our family registry have katakana-only names). When my mother was a child, she asked her mom how to write her name in Japanese. My grandmother wrote it down in katakana (ミドリ), telling her it meant either "green" or "beautiful bird." My mother wrote some letters to Japanese relatives and signed them in katakana. When she went to university, she took a Japanese course for the first time and learned about the existence of kanji; she felt terribly embarrassed, assuming that she had been signing her name incorrectly. She assumed that her mother had intended for her name to have a kanji but that she had simply not bothered to teach her how to write it (since katakana is easier to write). Only when she was about 60 years old she found out that her mother's name had only been in katakana and, therefore, her name is most likely to have been intended to be in only katakana: which was the prevalent custom in Japan at the time that my grandmother emigrated.
Since my mother gave me my middle name during those years in which she assumed she had a kanji but just didn't know which one it was, I take it that she wanted me to have a kanji. So when I started studying Japanese I chose 緑 for myself, since my favorite color has always been green. However, after I moved to Japan, the Japanese do not consider my middle name to have a kanji because my American passport does not have kanji in it, so they write my middle name as ミドリ as if it were a foreign word in all legal documents.
All that to say, you could choose to write your name as ミドリ in katakana. It is not a dumbing-down or kiddy way as my mother had thought, but rather a more historical way of writing female names, and if you immigrated to Japan someday, they would most likely not recognize your name as having any kanji.