The phrase "might I ask" is correct and can be used in written English.
It is often used when asking a question politely. For example, "Might I ask what the problem is?".
"Derek," said the great man, "might I ask you a very personal question".
"Might I ask you to refrain from throwing people down in the future?" "He also threw me down," Harris said.
Mrs. Clinton, while I find the prospect of a Trump Presidency to be terrifying, might I ask how you plan on winning over those Democratic voters who feel so upset by the information revealed in the D.N.C. hacks that they have difficulty trusting you?
Might I ask how much something like that cost?
Now might I ask one more thing?
What might have happened, I ask, if Clare and Plath had been similarly treated?
To get a sense of what might be, I asked Mr. Gregory to take me on a light-sensitive walking tour.
Curious as to what he might say, I asked Dr. Chapman for the one piece of advice he could give to me and my wife.
The controller might say: "I asked you a question.
Might there, I asked, be all-out civil war when the British and Americans leave?
Also, might I respectfully ask how you know that you're okay?
