How does Miyax measure the days?

This is an interesting and tough question. Readers are told throughout the story that it took a certain number of days for something to happen or that it had been a certain number of days since something previously did happen; however, keeping track of the days is an incredibly difficult thing for Miyax to do. The reason for this is her location. The story takes place in the Arctic, and for various parts of the year, the sun will not set. This means that Miyax does not have the luxury of a standard day-and-night cycle to judge a passage of days from.
As an alternative, Miyax is forced to judge the passage of time by combining other specific indicators. Her primary time keeper is the wolves. Readers are even told that specific days did not matter much to Miyax because she was mainly operating on "wolf time."
Although the clocks in Barrow would say it was time to get ready for bed, she was getting up, for she was on wolf time. Since there was no darkness to hamper her vision, night was as good a time to work as day, and much better if you were a wolf girl.
The operation and movements of the wolves became a rhythm of life for Miyax, and that rhythm is what mainly allows for Miyax to tell time and the passage of days.
Miyax did not know how long she slept, for midnight was almost as bright as noon and it was difficult to judge the passing of time. It did not matter, however; time in the Arctic was the rhythm of life.
Despite the fact that the sun might not ever set for months at a time, Earth's rotation in relation to the sun does cause the sky to slightly change color over the course of a 24-hour cycle. Readers are introduced to this concept in the first paragraph of the story. We are told that the sun was yellow in color against a lime green sky. We are then told that combination of colors signals that it is six in the evening. Presumably, six in the morning has a different sun and sky color combination. For longer periods of time, weeks and months, Miyax uses the sun's position relative to the horizon.

With a start, Miyax noticed the sun. It was halfway below the horizon. Shading her eyes, she watched it disappear completely. The sky turned navy blue, the clouds turned bright yellow, and twilight was upon the land. The sun had set. In a few weeks the land would be white with snow and in three months the long Arctic night that lasted for sixty-six days would darken the top of the world.

Miyax is so in tune with the sun's position that she actually uses it to figure out exactly which day of the year it is. That is quite impressive for a character so young.
About an hour later, the sun arose and marked the date for Miyax. It was August twenty-fourth, the day the North Star reached Barrow. Of this she was sure, for on that day the sun lingered below the horizon for about one hour. After that, the nights lengthened rapidly until November twenty-first, when the sun disappeared for the winter.