Traditional Reference

On August 8, 1974 Richard Nixon spoke to the nation from the Oval Office and announced that he would resign the office of president effective at noon on August 9, 1974.
 

If everyone knew, why did he need to write a letter to Henry Kissinger? And why Henry Kissinger?

Letter of resignation - Wikipedia, the free encyclopedia  (https://en.wikipedia.org/wiki/Letter_of_resignation)*

A formal letter with minimal expression of courtesy is then-President of the United States Richard Nixon's letter of resignation, under the terms of the United States Constitution. Delivered to then-Secretary of State Henry Kissinger on 9 August 1974, it read simply, "I hereby resign the Office of President of the United States." It was simply dated, but the recipient also recorded upon it the time of receipt, at which it took effect with important consequences under the Constitution.

Well, no. Not under the terms of the constitution, which only provides for the succession to the presidency should the president resign (or die or be removed).

*On the morning of September 21, 2018 the entry still contained the inaccurate information about Nixon's letter and the constitution.  Later in the day an edit corrected the error.  The new text reads:

A formal letter with minimal expression of courtesy is then-President of the United States Richard Nixon's letter of resignation, under the terms of a relatively unknown law passed by Congress March 1, 1792, likely drafted as the Constitution spelled out no direct procedure for how a president might resign. Delivered to then-Secretary of State Henry Kissinger on 9 August 1974, it read simply, "I hereby resign the Office of President of the United States." It was simply dated, but the recipient also recorded upon it the time of receipt, at which it took effect with important consequences under the Constitution.

Eyewitness To Power: The Essence of Leadership Nixon to Clinton, by David Gergen