What Kind of Expression Is This, Anyway?
Long ago you might have heard someone asking for a favor in this manner: “Would you be so kind as to fetch my shawl from the hall closet, dear. It’s a bit chilly today.”
In modern speech this formula has been abbreviated to “would you kindly, ” as in “would you kindly text me when you get there?”
In the shortened version it’s not obvious to some people who is supposed to be kind. The person speaking is asking the other person to do something kind.
When you scramble this expression by saying instead “may I kindly ask you to text me” you are calling yourself kind. It’s up to the other person to decide whether you are being kind in asking for a favor.
“I would like to kindly ask you to bring some flowers to the party” may seem polite at first glance, but the more logical version would by “Would you kindly bring flowers?”