It didn't take me long, here's the Tweet the OED sent out 2 weeks ago, on the occasion of JA's birthday:
I remembered having looked into that some years ago, and a quick check of my files revealed that I noted the following in May 2012 as I was looking at Jane Austen's Letter 52 dated June 17, 1808:
"JA wishes CEA “had not a disagreeable evening with Miss Austen (age 40) and her niece (must be still a girl). You know how interesting the purchase of a sponge-cake is to me.” That sounds to me like she is saying they are spongers, whose affections have been bought by a powerful person! So how fittingly ironic that a term JA coined to describe a person became a universal term for a kind of cake!...In the 1807 /Mirth and Metre, A collection of Songs Sonnets Ballads and Bagatelles /written by Charles Dibdin (and we knew his name as one that JA knew well!), on P. 125, in the midst, alas, of an anti semitic doggerel, I read:
“A pastry cook said, the Jew was a CAKE to give himself a bad character; but he found the world full of CAKES. He called a beau a SWEET CAKE; a lover, a HEART CAKE; a prude, a LEMON CAKE; and a wit, a SHORT CAKE; a doctor, a WORM CAKE, his patient, a BATH CAKE; a lawyer, a SPONGE CAKE, his client a PAN-CAKE; a courtier, a PUFF CAKE; a citizen, a PLUM-CAKE….”
So the timing is just amazing, obviously JA got the idea from that book."
[me again in the present] Yes, it surely is no coincidence that Dibdin's book is published in 1807, and then JA, less than a year later, writes a letter using the word sponge-cake in an anthropomorphic sense, EXACTLY as Dibdin coined a cake-term for a dozen different kinds of persons/cakes. Her satirical eye had been caught by his satirical lexicon of people as cakes, and she must have shared a laugh with Cassandra over it when they read it --- hence it made a convenient code for describing CEA's visit to their relatives who apparently were ripe for satirizing.
Here's the URL for the Dibdin book in Google Books so you can see the 1807 publication date plain as day at the beginning:
The doggerel about people-cakes is on ppg. 124-5.
And, again, this tells us that the OED is, as of two weeks ago, STILL not looking at Google Books in order to make their listings correct!