When Jesus declared that he was the one who was promised to come as the atoning Messiah, the Nephites would have immediately understood that Jehovah stood before them. Having established who he was, he turns to the covenants that he has made, declaring that they will be fulfilled. As God, he has the power to fulfill them.
Each of these verses references the Father. The Nephites have known their God by that title (such as 1 Nephi 22:9: “the making known of the covenants of the Father of heaven unto Abraham”). Jesus has declared himself to be Jehovah come to earth, and therefore the one who made the covenants. Therefore, he can promise to complete them.