This feels a little disjointed to me, since it makes little sense to admit ‘large voluptuous bottoms’. The next section clarifies what he is admitting (that they are very attractive on a baby penguin) but we don’t know this until we read it. It would make more sense to say “Large voluptuous bottoms are very attractive on a baby penguin,” Mayor Dinkleberry openly admitted.
But I agree with Irishman, the first example, no comma.
If you are going to split speech to clarify who is speaking and how they are speaking (and revealing intentions behind speaking), then you need to be careful what you write, since it must clarify what has happened previously if you are writing in past tense.
“What’s the difference between a computer and a blonde?” Gary laughed, thoroughly pleased with his repertoire of corny gags. “The computer is smarter, but the blonde is easier to turn on!”
Whilst we know Gary finds himself hilarious, it feels slightly odd that we know this before the punchline so to speak.
As others have stated, the use of the word that indicates that you are paraphrasing and not quoting. Drop the comma after that, delete the quotation marks, and do not capitalize the “L” in “Large.” Or, if it is a direct quote, delete that, use the comma, and capitalize.