This is a terrific article from a couple of years ago suggesting that the Doctor needs to meet Jane, and reminds us that she created the modern English novel (throw that at the next person who scoffs at her as a chick lit writer!): Jane Austen Should Be On Doctor Who
Much as I would love to see this myself (see previous post), it cannot be with Mr. Smith, darling as he is. He's cute, he's silly, he's surprising, but he's not sexy (not like Tennant and Eccleston) or very good at flirtation.
She'd think of him as an amusing, scrappy little thing, probably dig his mind, but I seriously doubt there'd be good chemistry between them. Matt needs some protein shakes and some dumbells to start. Tennant would be the best choice, but as he's lost to us now, we can dream of a future Doctor meeting up with our brilliant Jane.
If you can't wait, and if you like audio episodes, here is one from Big Finish where the Doctor does meet Austen: Austen/Doctor Who Big Finish Audio.
I have a few audio episodes myself, but this doesn't mean they shouldn't do a TV episode!