It feels like one of the deadly sins but it's actually a physiological need...
If it's a physiological need because you were sick at the time of the exam, then your school should have a policy about how such situations are to be handled.
While I sympathize with your plight, it sounds like the problem was that you were too tired and just slept through the exam. If that is the case, then I'm afraid to say that the professor may have every right to assign you a zero for not taking the exam. There are some schools where the policy would be to assign an incomplete in such circumstances, but this is entirely school-dependent.
The essential issue is that if you do not have an excused absence for an exam, you're at the mercy of the instructor and any existing policies. At my school, for instance, the policy says that anything other than an approved university activity cleared in advance is at my discretion. So if the instructor chooses to give you a zero, there's nothing you can do.
You also need to consider things from the instructor's point of view: giving you a do-over because you slept through the exam would require that she give any future students who did the same the opportunity for a do-over as well—after all, why should only you get special treatment? Granting your request would mean that, in the future, all you'd need is to say "sorry, I overslept" and you could postpone an exam you weren't ready for. This is simply unfair to the instructor and to all the students who prepped for the exam at the scheduled time. This is simply not a fight you can possibly expect to win unless there are exceptional attenuating circumstances that have not been mentioned.