Shalom and thank you for your question! You wish to know when the Haftorah of Isaiah was included in the Yom Kippur services. I do not know the exact date, but will share some background with you.
Firstly, Yom Kippur is a day of atonement, and you can read that as at-one-ment. Returning to be one with our Creator, after having sinned with the Golden Calf. Many details of its institution and observance are recorded in Leviticus. It remains till today the most solemn and spiritual day of the year, truly a day for deepening our connection with G-d.
When various foreign rulers (such as the Greeks and the Assyrians,) ruled over Israel, they forbade public reading of the Torah, in an attempt to wipe out devoted adherence to Jewish belief. The Sages, left with little choice, instituted the reading of sections of the prophets, know today as the Haftorahs, (or Haphtorahs.) These sections were chosen for their implied connections with the content of the Torah readings from the Chumash that they were temporarily replacing. This served the purpose of keeping up connection with the Torah portions, so that they should not be forgotten. The custom remained even when the threat was no longer relevant.
There have also been many people over the ages who deny that the writings of the prophets are a part of the Torah. Reading the Haftorahs as part of the prayer service shows that these writings are just as holy. In fact, the Sages (of the Talmud) teach us that “the words of the Sages are more cherished than the words of the Torah (Chumash,)”
which is not meant to lessen the holiness and importance of the Chumash, but to point out a hidden truth.
Chassidic teaching explains that there are hidden levels of G-dly light in the writings of the Prophets.” G-d desired a ‘dwelling place’ in the lower worlds”. In other words, G-d wishes us to take the truth of Torah belief and apply it to our lives, including our trials and tribulations, in every generation. The Prophets, as well as the righteous leaders of each generation, have the ability to do this, and to help us do this also.
Rabbi Shalom Dov Ber of Lubavitch, for example, the Fifth Lubavitcher Rebbe, was a widely acknowledged Torah scholar, and well versed in every aspect of the Torah. Nevertheless he was known to take exceptional care to prepare for the reading of the Haftorah.
The reading from Isaiah talks about clearing the way of obstacles from the Jewish people, in order that they be able to overcome temptation and come close to G-d.
This is the ‘why’ of your question, if not the when…
