I think you are mixing up tag questions.
It's your book, isn't it?
with a construction like
Let him speak, will you?
This latter is a rhetorical re-ordering of "Will you let him speak?"
It is pretty unusual to use "tag questions" with the imperative. It is possible to have
Go to the shops, why don't you?
but you can't add a tag question
Go to the shops, don't you? / This one doesn't make sense.
So coming back to your phrase, you could have
Let them wreak havoc again, did you? / Let them wreak havoc again, shall we?
as rhetorical re-arrangements of
Did you let them wreak havoc again? / Shall we let them wreak havoc again?
But you can't add a "tag question" as in your example. If you are not sure about the rhetorical effect of the re-arrangement you should be careful. It is sarcastic
Husband! What is going on here? You're meant to be watching the kids but there's a mess all over the living room. Let them wreak havoc again, did you?? And who's going to have to clear up? I'm really tired of your lack of responsibility.