This is a sentence in the form of
as ... as
Sentences in this form commonly denote a comparison. What President Obama is saying is that his wife Michelle is as talented as he knows she is, meaning that the President believes that Michelle is very talented. And due to the fact that Mr.Obama thinks that Mrs.Obama is talented, Michelle is talented according to the quote. Do note, however, that the comparison could go both ways. Obviously, the President was complimenting his wife so he is trying to say that because he knows Michelle is talented, Michelle is thereby talented. But if one were to use the quote as an insult, in the case that one believes Michelle is not talented, then one could claim that Michelle is not talented (this is more of an aside to explain how sentences in the form as ... as can be used).
If you believe that the wording "as talented a person" sounds awkward, then I'd like to reference the name of an article that uses something similar to this wording,
John Wall shows he’s as great a person as he is an athlete
John Wall is a professional basketball player in the United States, so he is clearly an athletic person. The statement above is trying to say that his quality of being of a good person is equatable to his quality of being a good athlete. Since John is a good athlete, he is a good person. It could be argued that the meaning of these kinds of sentences require knowledge of the stuff after the second as. For example, if you don't know whether or not President Obama believes that Michelle is talented, then you can not infer much from his quote. And if you did not know that John Wall was a professional athlete, you would probably be confused by the statement I referenced above.
The reasons why your other two proposed sentences are incorrect are that the first one sounds awkward (as talented of a person as) and the second one uses three instances of the word "as" (as talented as a person as I know).
EDIT: So after looking up more sources about this specific quote, I now realize that I was wrong in my original answer and that Martin Smith's answer is correct. The quote in question was a response to an inquiry by Rolling Stone about the possibility of his wife, Michelle, running for office. It turned out that the president said something similar on a radio show called Sway in the Morning,
She is as talented and brilliant a person as there is, and I could not be prouder of her.
Clearly, President Obama is trying to convey the same message in both answers to the same question. That her wife is the most talented person he knows. So thank you to all the users who gave me the feedback that allowed me to come to this realization.