Sunagakure had directly announced its surrender without resistance.
The Daimyo of the Land of Wind naturally couldn't still choose to foolishly resist.
Almost in coordination with Sunagakure, just as the village surrendered, the Daimyo of the Land of Wind made a public appearance. Together with his "good neighbor," the Daimyo of the Land of Earth, he officially declared their submission to the Snow Empire through official channels. They would both step down from their positions as daimyo on the day of the announcement, acknowledge the authority of the Daimyo of the Land of Snow, Kazahana Koyuki, and honor her as their one and only shared sovereign!
This was the last bit of face left for these daimyo.
Even in surrender, they could choose to submit to Kazahana Koyuki, a fellow daimyo, rather than anyone else. The Snow Ninja Village naturally wouldn't quibble over this, and Naruto had no intention of changing the system of nations, daimyo, ninja villages, and Kage. Every world had its own particularities. At least in this world, the arrangement of nations and ninja villages was already fairly well established, so it couldn't be changed rashly. Nor did Naruto's upcoming reform plans include that. Of course, that was only true of the overall structure. Naruto still intended to make changes to certain necessary details; otherwise, the Guardian Shinobi Twelve incident in the Land of Fire from the original story would be the perfect cautionary example.
Within a nation,
who held the greater authority, the daimyo or the Kage?
If that issue couldn't be resolved in a reasonable and lawful manner,
Naruto and Kazahana Koyuki's generation wouldn't have any problems. But if the Snow Empire lasted for several centuries—not that it would even take that long—once their generation had passed away, turmoil would inevitably erupt within the empire.