Here we have a very curious statement that comes so simply that it is easy to miss. To highlight the modern improbability of this, let us restate what has just happened.
Ammon has come among a foreign and hostile people armed for hunting, but certainly armed. Ammon is taken as a ritual captive, and his complete humiliation and subordination to the king of Ismael is demonstrated by his public binding, and being carried before the king. For political reasons alone, this captive might be held in prison (and no doubt publicly tortured, if we use the Maya model) and possibly killed. Again with the Maya model, this is not an execution, but probably a public sacrifice, making a religio-political statement rather than simply extinguishing the life of an enemy.
With such a dire future, Ammon says that he wants to live with the Lamanites, and the king says, in effect, "oh, what a good idea. While you are at it, why don't you marry my daughter?" Doesn't something seem rather wrong about this?
Absolutely not. In fact, in the context of this type of society, the king's response is actually almost the only one that would allow Ammon to live. In kin based societies, such as we find in Mesoamerica, and such as are evidenced in the Book of Mormon, marriages create nearly unbreakable links between families. We are familiar with political marriages in Western royalty, but this is even more important, and much more binding.
Let's return to the problem as King Lamoni would see it. He has a man who has voluntarily come among his enemies. This man, while presented as a captive, is not a captive of war, but has come willingly. Thus there is little political and religious significance in his death or captive-torture. This man has professed that he wants to live among the Lamanites, but it is possible that he could want to do so to be a spy. Thus there is a problem facing the king. He could kill him, but achieve very little, or he could let him live, but worry about what he might do in the future. What the king decides to do is allow him to live, and morally and legally bind him to the Lamanites by marriage. This is no simple marriage, but one into the royal family. Thus the king allows Ammon to live, but places him in a position where his allegiances must change (to local family rather than the distant Nephties) and where the king might be able to keep an eye on him as one of the retainers of the king. What appears to be incongruous, is consistent in context.