As Nephi likened Isaiah 49 to himself and his people, it would have become an irresistible confirmation that Jehovah watched over the Nephites. These verses, in particular, would have been seen as directly relevant.
While Isaiah was the prophet to whom Jehovah spoke in the Old World, Nephi saw himself as a parallel prophet in the New World. Isaiah might be a servant to raise up the tribes of Jacob, but Nephi was the servant to restore at least some of the scattered tribes of Israel.
In particular, the prophet is given as a “light to the Gentiles, that thou mayest be my salvation unto the ends of the earth.” Isaiah would have seen this prophecy in a different light than did Nephi. When Nephi is writing these words, he is the leader of the Nephite people in the New World. The only way that the population of the Nephites could have risen quickly enough to build a city at this early stage is to have merged with some of the indigenous peoples who are known to have been in the New World when they arrived. Even without defining precisely where they arrived, they could not have arrived on an habitable coastline that was not already inhabited, according to known archaeology.
Thus, for Nephi, the gentiles were already among them, and Nephi teaching them the gospel became a light for them unto their salvation. In the New World, Nephi would have seen the prophesy of kings and princes worshipping Jehovah as a prophecy of their success in spreading the gospel in the New World.