Are you not already being kind by saying thank you to begin with? I read the comments on this site What's the deal with thank you kindly?, and I guess that I am still pretty confused as to the meaning of this.
Obviously, if someone says "Thank you" in a very sarcastic manner then they aren't intending to be kind, but I mean in the general sense of showing gratitude to someone for something, is there a way to do this unkindly? If there isn't a way, then why would someone say, "Thank you kindly"?