meaning - "I kindly ask to" vs "I ask to kindly"

Incorporating caesarsgrunt's suggestion:

Would you be so kind as to send the letter to your boss?

If you would be so kind, send the letter to your boss.

The speaker is very politely asking the listener to do a menial task. This is very formal speech and might sound "stuffy", "pompous" or even worse, "patronising" to some. Very much depends on tone.

Kindly, send the letter to your boss

Again the speaker is asking for a favour but it is less formal. It means I am asking you kindly to do something for me.

However in a British context, it would be more appropriate to hear the follow:

If you wouldn't mind sending the letter to your manager.

Would you mind sending the letter to your senior manager?

Oi, you! Send this off to the boss. (Very informal!)

EDIT: I've added the more formal expressions, manager and senior manager as suggested by Tim Lymington.