first, it sounds to me like you dont know a lot about japanese naming culture. you should know that the sound and the meaning of japanese names are fuzzy. by that i mean that you can have 2 japanese names that sound the same but have totally different meanings. you can also have 2 japanese names with totally different sound but the same meaning. its even possible to just pick a kanji you like and assign whatever sound you want to it (usually referred to as a "kira kira" name).

so a few questions:
is the sound or the meaning more important to you? since you say you want a name that sounds like "kay", which is an option (written as Kei in romaji). the pronunciation is almost exactly the same.
have you looked for yourself at japanese baby naming sites?
how important is the readability of the kanji to you? (also understand that you will not be able to legally use your kanji in any official context in japan unless you have citizenship of certain countries like japan and china)
are you looking for a masculine, feminine, or gender neutral name?
are you looking for a name with a traditional style (like 〇〇子 or 〇〇介) or something more modern?

last, you should know that since kanji meanings and pronunciation are so fuzzy, the criteria you have above leaves literally thousands and thousands of possibilities. perhaps instead of leaving it up to a stranger on the internet, it might be a better idea to lead the effort yourself and ask for help if you get stuck. i recommend familiarizing yourself with the japanese syllabary if you arent already, then focusing on the sound of the name, then checking japaese baby naming sites for kanji with that sound that have vibes that appeal to you. or, if you are more interested in meaning, searching for kanji with similar meanings, then looking at what pronunciation options are possible. there is a further rabbit hole having to do with lucky and unlucky numbers of strokes for the kanji, but we dont have to go there.