Rabbi Chaim Friedlander zt”l in the Rinas Chaim writes about the issue of Kingship after Rosh Hoshana:
“The similarity of issues, which appear in the prayers of both Rosh Hashanah and Yom Kippur, pointing to Hashem Yisbarach’s kingship and reign, leads us to a question. Why must we bring up the issue of malchus, Hashem’s kingship, on Yom Kippur as well?
Rosh Hashanah is the Day of Judgment, and the whole concept of judgment is a product of His kingship. Hashem Yisbarach, as the supreme monarch, distributes tasks — and the vehicles necessary for the fulfillment of those individual tasks — to each one of His subjects on Rosh Hashanah. Thus, on the first day of the year, HaKadosh Baruch Hu dons the cloak of the supreme Judge and estimates the quality of each person’s fulfillment of his tasks from the previous year. Those individual tasks are part of the general goal of proclaiming Hashem Yisbarach as King over creation, and over each one of us in particular. Hashem Yisbarach then delegates each person’s task for the coming year according to the level of his performance the year before.
However, due to Hashem’s lovingkindness, the judgment does not end on Rosh Hashanah, but lasts during the subsequent Ten Days of Repentance, during which it is still possible to repent and to amend the final verdict. On each of those ten days we en treat HaKadosh Baruch Hu with the supplications of “Inscribe us in the Book of the Living,” and “In the Book of Life… may we be inscribed before You.”
The whole issue of judgment is maintained within the concept of kingship, as we stated before. We are judged according to what extent we have accepted upon ourselves Hashem Yisbarach’s kingdom in all aspects of our lives, and especially in the fulfillment of our individual tasks. Our judgment also hinges upon the extent that we are prepared spiritually for the holy task of proclaiming Him as King in the forthcoming year. That is why we stress kingship in our prayers during those ten days, saying “the holy King,” and “the King of judgment.” All these ten days are days meant for us to proclaim Hashem as King over us — and our judgment flows from this.
The conclusion of the judgment occurs on Yom Kippur. On Yom Kippur it is assessed and established to what extent we are spiritually ready to recognize the reign of our King, the King of all kings — HaKadosh Baruch Hu. Therefore, on Yom Kippur we mention and we seek the acceptance of Malchus Shmayim, the Heavenly kingdom, just as we do on Rosh Hashanah. On Yom Kippur the spiritual task of the entire ten days of proclaiming Hashem King comes to its peak and culminates with the acceptance of Ol Malchus Shmayim at the end of the Ne’ilah prayer.”