a subtle difference
“Just a kind reminder” infers that the reminder itself is a kind one. Kind being an adjective describing reminder.
“Just a kindly reminder” infers that a reminder is being given kindly. Kindly is an adverb describing how it is being given.
I think we would find that in some cases the first sentence is being used incorrectly where they actually mean the second second sentence.
https://www.englishforums.com/English/JustAKindKindlyReminder/hqvbr/post.htm
My Diary - Leonardo Cardillo 