Kanji isn't used in only formal situations, and that is the proper way to write kawaii. If you look at the actual kanji, it would read kaaii, but since that's the proper way of writing it even google translate will decode it properly to kawaii -> cute.
I know kanji =/= formal, but 9.9/10 times, kawaii by itself will be in hiragana. Kawaii is a very casual/informal word, so nobody ever uses it in kanji, you just don't use it like that. If you were IMing a Japanese person and you used kawaii in kanji, they'd laugh at you and say "流石外人ですね." However, if you were using kawaii as in 可愛がる, which can be used in formal situations, it would be in kanji.
Japanese have rule breakers (just like English) all the time and kawaii is one of them. It's really weird trying to say "Ka-aii" (you'd have to say it with two long vowel sounds in a row, can't pronounce it like 会), so they threw a W in there. That sort of pronunciation shows up in a lot of other places.
