"Thank you so much for your kind birthday wishes" is "생일 축하해줘서 고맙습니다." in Korean.
When we wishes someone happy birhday, we usually say "생일 축하드려요"(to not really close people) or "생일 축하해"(to very close friends or younger people) or "생신 축하드려요"(to very old people with respect).
We sometimes says to very old people "오래오래 건강하게 사세요!", which means "I wish you healthy and long life."
Of course, a dating couple may speak sweet words other than above things, but they are too diverse to mention.