To me the sentence: "To whom, might I ask, am I speaking?" is a changed word order of "Might I ask to whom am I speaking?", and it's not grammatical.
I think it should be: "Might I ask to whom I'm speaking?"
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As Stangdon has aptly put it, "Might I ask" is an interjection which can more succinctly be called a parenthetic clause set off from the main sentence by intonation in speech and by comas or the likes in writing—explanatory or qualifying clause not essential to the main construction. There are two inversions— one of 'might' which is just rhetorical only to add flavour and the other is the inversion of 'am' for questioning. The former is optional but the latter is a necessity.
If 'whom' is used, it is better that 'to' be prefixed to it. But if 'who' is preferred leave 'to' at the tail end. So first two sentences are quite right.
The suggested line can only be accepted if we just view inversion of "Might" for questioning (not rhetorical) bereft of quotation.
We fail to agree in this respect.
Your version and the original version are both correct. English allows some flexibility in word order.
The original is more colloquial and you're unlikely to find it in a formal text. Your edit is more formal.
To expand a bit on the other answers, I would say that both sound equally "correct". However, the first sentence,
To whom, might I ask, am I speaking?
Puts the emphasis on "to whom", and almost sounds confrontational.
Might I ask to whom I'm speaking?
Sounds more like a polite request.
I should say though, that both examples sound incredibly formal, and almost no one talks that way in modern english unless they want to sound posh or very passive aggressive.
Maybe you're not concerned with that and just curious about word order, but just in case you or someone else comes across this and wonders how an average person might say this, since as it's written it sounds like either a fantasy novel where some haughty palace guard is trying to keep someone out and be annoying at the same time, or some inexplicably formal desk clerk that's answering a phone.
Assuming it's a situation where you pick up the phone and would like to ask who's calling, you might say
Hello, who's this?
Can I ask who's calling?
or if you want to sound extra polite
May I ask who this is?