Of course, the phrasing "This is what I shall explain..." implies that Cauchy has just stated what this theorem is, so it would seem that yes, we should have a very good chance at finding out what it was, provided we can find out where the statement is from.
It appears to be from a note published on the 4th of May 1857 called Sur l'utilisation des régulateurs en Astronomie. This is the last article in the last volume of "Série 1" of Cauchy's collected Oeuvres, excluding what appears to be the minutes of Académie on the day Cauchy's death was announced, and assuming the date under the article's title is the publication date, it appeared just a few weeks before Cauchy's death. Thus it appears likely that this was indeed his last publication.
It's article 589 in Tome XII, Série 1 of Œuvres Complètes d'Augustin Cauchy, page 455, an online copy of which can be found here.
The note seems to be about the use "Régulateurs" in astronomy, where I assume a "Régulateur" is an astronomical regulator, an old type of pendulum clock used in observatories.
Translation (my own, and although I'm fluent in french I'm less fluent in 19th century astronomy, and without knowing what exactly Cauchy is talking about I have no idea what the correct english technical terms are - I've therefore simply translated them as literally as possible):
On the use of regulators in Astronomy
I noted in the previous session the advantages of using regulators in mathematical Analysis. I will add that not only the variables, but indeed the parameters contained in the given equations, finite or differential, or even partial differential, can be supposed to be developped according to the ascending powers of a given regulator. In many problems, particularly in Astronomy, this observation will allow us to render monodrome or monogeneous the variations of the various orders of unknowns which are expanded as series following the increasing powers of a single given regulator. This solves the question raised in my previous Mémoire, on the possibility of developping the coordinates which determine the orbits of the planets about the Sun, or of satellites about the planets, according to the ascending and descending powers of the trigonometric exponentials whose arguments are the excentric or average anomalies, and thereby according to the ascending or descending powers powers of the keys of the orbits. This I will explain in more detail in a future Mémoire.
I frankly have no idea what any of the above means. The article is followed by the aforementioned bit of text about Cauchy's death.