In a statement such as "Kindly answer my letter soon," one with equal correctness and propriety could write "Please answer...." In such use, kindly and please are courteous formalities that convey a primary meaning of "obligingly." Each, a polite addition to a request, no more conveys exact meaning than the dear in the salutation of a letter expresses affection. Please is the more commonly used word in this situation, possibly because kindly seems to some people to carry an air of affectation or smugness. Take your pick.
Dictionary of problem words and expressions. Harry Shaw. 1975.