Clearly the Korban Pesach Sheni was consumed on the night of Iyyor 15th, hence that many people consume a piece of Matzo on nite of Iyyor 15th rather than afternoon of Iyyor 14th.
14-15 of the month may possibly be connected to the fullness of the moon.
Needless to say, penultimately the answer is that this is a Gezeiras haKosuv to be an entire month later.
Also, please bear in mind, that the original Divine Plan, was to have a Yom Tov, each month during the Summer. Pesach Sheni is the Yom Tov for Iyyor. Came Tammuz which should have had the grandest of all Yomim Tovim, and got messed up by the Eigel haZohov, striking 3 months with no Yomin Tovim. ….. These 3 holidays are all made up to some extent in Tishri, hence 4 separate holidays in Tishri:- 1) R.H. 2) Y.K. 3) Sukkos 4) Shemini Hag ho’Atzeres to pay back Tammuz, Ov , Ellul as well as have one for Tishri itself.