Some possible candidates--
Jane Yolen's "The Emperor and the Kite"The Kite Mystery, by M. Adrian (1968)
Nu Dang and His Kite, by Jacqueline Ayer (1958 publication year)
The Dragon Kite, by Thomas Lewis (1973)
A Sky Full of Dragons, by Mildred Whatley Wright (1969) <--- I think that this MAY be the book that I was thinking about above, but no longer as sure that it's the one whose plot I recall matching yours.
Sumi's Prize, by Yoshiko Uchida (1963)
As I noted above, it's possible that I'm conflating the illustrations in another book that I remember on this topic-- it was one that my mom had a LOT of books about-- and living on the west coast, it was very relatable culturally sensitive literature for her elementary students, so I've read a LOT of K through 6 books about Asian culture and heritage and kites in that context. I do recall the book that you're mentioning though-- at least I think that I do!!