Marianne gave us a sample of Jane Austen writing more or less casually to her sister: see this comment on the Emma post. Here's a somewhat more formal letter, from The Oxford Book of Literary Anecdotes, a book which I bought back in the '70s and which has given me in enjoyment at least 100 times the worth of the original cover price of $1.75. 

From 1811 to 1820 the son of George III ruled as Prince Regent. Jane Austen sent him a copy of Emma, and his secretary replied, on behalf of the prince, that he had very much enjoyed it, and added that Austen might wish to dedicate a future work to the prince, and moreover that "any historical romance, illustrative of the history of the august House of Coburg, would just now be very interesting." It isn't clear (to me anyway) whether that suggestion came from the prince or the secretary. 

Austen replied:

To James Stainier Clarke, 1 April 1816
My dear Sir,

I am honoured by the Prince's thanks and very much obliged to yourself for the kind manner in which you mention the work.... You are very kind in your hints as to the sort of composition which might recommend me at present, and I am fully sensible that an historical romance, founded on the House of Saxe Coburg, might be much more to the purpose of profit or popularity than such pictures of domestic life in country villages as I deal in. But I could not more write a romance than an epic poem. I could not sit seriously down to write a serious romance under any other motive than to save my life, and if it were indispensable for me to keep it up and never relax into laughing at myself or other people, I am sure I should be hung before I had finished the first chapter. No, I must keep to my own style and go my own way, and though I may never succeed again in that, I am convinced that I should totally fail in any other.

I remain, my dear Sir,
Your very much obliged and sincere friend,

J. Austen

A minor aspect of the amusement in this, for me at least, is that I regard Austen, as far as I know her, as having provided the prototype for the modern "romance" genre, the word of course having had a very different sense in Austen's time. This is not a criticism of her, but rather evidence that the general outline of that plot (if my limited acquaintance with it can be  validly generalized) has an archetypal sort of appeal: man and woman who initially dislike each other but by a gradual opening of the eyes are drawn together by various desirable qualities in each other (noble in Austen, probably often less so in our day). The preferred end is a wedding. This broad outline is as simple as that of the murder mystery, and seems to be as enduringly popular.