B”H
Q. Shalom Dear Rabbi.
Why didn’t King Saul kill the Amalekite King immediately?
Is there any Midrash or other source that gives specific, detailed information about the conversation/exchange between King Saul and the king of Amalek before Prophet Samuel killed the Amalekite king?
Thanks!

A. There are a number of reasons quoted in Meforshim as to why Shaul Hamelech didn’t slay Agag immediately, to mention a few:
Talmud (Yuma 22b) teaches that Shaul created a kal vachomer from eglah arufa. Namely if Hashem has compassion on one single murdered plain victim, so much more on the many and on a king.
Rav Tzadok Hakohen zt’l (Likutey Maamorim 64, 65) mentions that Shaul was aware that the descendants of Agag would eventually teach Torah to many, King Shaul maintained that for the sake of Heaven and the greater good, one may even transgress temporarily Hashem’s will.
Orah Vesimcha (p.20) explains that Shaul Hamelech foresaw that by temporarily staying Agag’s execution, the most important willing and joyful, second Kabbolas Hatorah of Purim will come to be.
Ahavas Yonosson, clarifies that Shaul maintained that only Agag’s mother was from Amolek and not his father, he therefore was exempt from the extermination decree.
Daf Al Hadaf (Yevamos 46b) quotes the Rogitchover Gaon zt”l saying that surprisingly Agag converted to Judaism to save his life. Once a ger, Shaul could not execute him. Shmuel Hanavi, disagreed and asserted that you don’t accept gerim from Amalek even after the fact.
Horav Shlomo Miller’s Shlit’a opinion is that Shaul Hamelech wanted to execute Agag publically to publicize to all the victory and to openly show Hashem’s justice, thus creating a great Kidush Hashem. (See also Yikra Deuraissa p. 12).

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a