"kindly explain" is correct and usable in written English.
You can use it when you are asking someone to explain something to you in a polite way. Example: "I'm not sure I understand. Could you kindly explain what you mean?".
M. HERZOG: Would you kindly explain to the Tribunal under what circumstances you were appointed to that office?
So would you kindly explain the difference that I perceive in both the statement versus the way you just answered?
"They kindly explain, 'We are insulting you in case you get a big head.' It's pre-emptive slagging". "Artemis ruined everything," says Donal. "Eoin and I used to have a little theatre company going.
I'll tell you, sometimes it has seemed to me that you were looking back through life, back through troubles I pray you'll never have, asking me to kindly explain myself.
Germaine, if Hirst's spot paintings are "triumphantly vacuous" and painted by somebody else, kindly explain why the Government Art Collection should spend public money (£170,000) on such a thing?
If your roommate pops their head around the door and tells you to stop being a bore and come to the kitchen for a few drinks, kindly explain to them that your essay is far more important than hanging out with friends... Fat chance.
But he very kindly explained what he meant by that in the margins.
Take those in Serie A, for instance, as our intrepid correspondent James Richardson kindly explains.
Griffin duly buttonholes Parker (who kindly explains that she is wearing a dress by Chanel), while Christopher Plummer (Oscar-nominated for The Last Station) stands obediently to one side.
She kindly explained that it took more than a good voice to become a singer and that some things just weren't possible.
She reads it, frowns, calls Bachman at 2am to ask "What's this poem about then?", and, once he's kindly explained, decides there's time for a few good nights before that one.
