What a great question.
>> Who am I speaking to?
(This is grammatically dodgy.)
It should be:
>> Whom am I speaking to?
(But, this is a bit pretentious.)
>> To whom am I speaking?
(This is grammatically sound but it's very pretentious.)
So, the choice is: Do you try to sound natural or do you speak correctly?
I'm afraid it's not for me to answer that. Personally, I would say "Who am I speaking to?"
I know it should be whom and I know I'm ending a sentence in a preposition, but getting those wrong is better than the "fix" if you ask me. The fix sounds awful.
Can I ask who's speaking?
Good question. It's not often I recommend deliberately getting the grammar wrong!
Here's a lesson on who and whom.