Short version: consider Savage's theorem (fulfilling the conditions by offering Sleeping Beauty a bet along with the question "what day is it?", or by having Sleeping Beauty want to watch a sports game on Monday specifically, etc.). Savages theorem requires your agent to have a preference ordering over outcomes, and have things it can do that lead to different outcomes depending on the state of the world (events), and it states that consistent agents have probabilities over the state of the world.
On both days, Sleeping Beauty satisfies these desiderata. She would prefer to win the bet (or watch the big game), and her actions lead to different outcomes depending on what day it is. She therefore assigns a probability to what day it is.
We do this too - it is physically possible that we've been duplicated, and yet we continue to assign probabilities to what day it is (or whether our favorite sports will be there when we turn on the TV) like normal people, rather than noticing that it is meaningless since we might be simultaneously in different days.
I disagree with the part "her actions lead to different outcomes depending on what day it is." The way I see it, the "outcome" is the state of the entire multiverse. It doesn't depend on "what day it is" since "it" is undefined. The sleeping beauty's action simultaneously affects the multiverse through several "points of interaction" which are located in different days.
Vladimir Slepnev (aka cousin_it) gives a popular introduction to logical counterfactuals and modal updateless decision theory in the Tel Aviv LessWrong meetup.
[https://www.youtube.com/watch?v=Ad30JlVh4dM&feature=youtu.be]