Would you use a comma or question mark (or both) between the words “speeding” and “an” in this sentence?

When asked by a young patrol officer "Do you know you were speeding?" an 83-year-old woman driver talked herself out of a ticket by stating, "Yes, but I had to get there before I forgot where I was going."

I would solely use the ? and not a comma.

Just the question mark, but you do need a comma after 'patrol officer'.

I would include the question mark AND the comma after the quote. This is where the comma would be without a direct quotation, i.e. "When asked by a patrol officer, an 83-year-old...". You could always remove the direct quote and write it "when asked if she knew she was speeding", but that's just an alternative if you're unsure about the commas.

When asked by a young patrol officer "Do you know you were speeding?" is a clause that can be dropped from the construction and the remainder "An 83-year-old woman driver talked herself out of a ticket by stating, "Yes, but I had to get there before I forgot where I was going."" can stand on its own. Clauses that can be separated should be set off by a comma. It is also lingually used as a break in the connection of "...you were verbing an object..." No need of the hyphens in 83 year old.
The sentence might be constructed more easily by:
A young patrol officer asked an eighty-three year old woman if she knew she was speeding and she replied. "Yes, but I had to get there before I forgot where I was going."

I disagree; I would hyphenate an age when it is being used as an adjective.

It can be done, but please note I said no need (the hyphen is required when spelling out the age number as I did). I didn't say wrong. It's a matter of choice it seems in that hyphen application. And the whole posed posting is a matter of style too. In a joke relation environment, the structure passes as is and in all the other ways suggested here. They all achieve the result of getting the point across and that is the basic premise of language. It does have the several noted failings too.
Researching punctuation rules (not grammar which has to do with what form of word to use) I stumbled upon this item in Wikipedia under "Punctuation":
Novel punctuation marks An international patent application was filed, and published in 1992 under WO number WO9219458,[11] for two new punctuation marks: the "question comma" and the "exclamation comma". The patent application entered into national phase exclusively with Canada, advertised as lapsing in Australia on 27 January 1994[12] and in Canada on 6 November 1995.
http://en.wikipedia.org/wiki/Punctuation
And for a guide to punctuation rules (which change as language changes, as evidenced by the patent above) try this
"Guide to Punctuation" by Larry Trask which can be found linked at:
http://www.informatics.sussex.ac.uk/department/docs/punctuation/node00.html