A determiner answers the question "which?"
Since a possessive in front of a noun answers that question, it assumes the function of a determiner and normally "takes the place" of an article, etc.
I took the ball - I took Sally's ball (article not needed).
When adjectives are involved, the determiner comes first, then the adjectives, then the noun.
I took the red ball - I took Sally's red ball.
But the premise of your question is wrong. Because this really does not make sense:
I took fifteen minutes' break.
It does not make sense for "break" to "belong" to anything called "fifteen minutes"
However, "fifteen minutes" (plural, not possessive) can modify break to describe the type of break. And you would use a determiner/article as you would without the modifier.
I took a fifteen minutes break.
Even so, this sounds better:
I took a fifteen minute break.