King David did not sleep at night because he did not want to taste death, so he developed a system, proper calculations on how to accomplish this. From what it looks like Vilna Gaon developed a similar method. Do they write anywere in detail on how to do this?

The Talmud [Sukka 26b] states that King David would not sleep more than 60 breaths. Even though it is meantiond in Shulchan Aruch [OC 4; 16] the commentators [see Mishna Brurah] explain it is only "at a time" but he slept several units of sixty breaths. The Biur Halach [ibid] discusses the time in minutes these sixty breaths took, the opinions diverse form 3 minutes up to 3 hours. In any event it is not a recommended practice for the lay person and does not apply on Shabbat As for the Vilna Gaon, there is a story he slept 4 hours a day and he related it to a Pasuk [see Responsa Yabia Omer 4 OC 2]