The Talmud explains the arrangement of the Amidah benedictions in Megillah 17-18. The rationale for placing the prayer for prayer after the prayer for David is given there as follows:
וכיון שבא דוד באתה תפלה שנאמר והביאותים אל הר קדשי ושמחתים בבית תפלתי
The prooftext quoted (Isaiah 56.7) does indeed mention prayer, but it says nothing about David. The reader is left wondering how the prooftext proves the positioning of the prayer for prayer.