This. A thousand times this.
Are there relatively few black Muslim women who apply to Duke? yes. So, the facts alone will get you noticed - but on its own, that won’t make you stand out as an applicant. Neither will having been hospitalized in a third world country (also, that is not as rare as you might think).
The Dean of Admissions at Princeton once said that what they look for is what you have done- and are doing- with whatever experience you talk about. So, in your example, you experienced being hospitalized in a third-world country. Presumably you had some realizations about the differences between your US world and that world, which made you more appreciative of what you have and more determined to make a difference. If so, what have you done to act on that already? What does that show about you as a student and a future citizen?
Similarly: being a black muslim woman is a fact, and if you are living in the US pretty certainly has extra challenges. If you write an essay on how those challenges have affected you, and how you are navigating those challenges, make sure to push yourself to the next level of thinking as your frame your essay. Link the past to the future.
Remember that goal of the essay is for the AO to think ‘this is somebody who brings something we need to our community’. Look at your application as a whole: every part of it adds to the overall picture of you: your grades and test scores show that you can handle the work; your ECs speak to how what you add as a member of the community, and your LoRs show your teacher’s & GC’s perspective on who you are. Your essays are your chance to knit all those pieces together to show the AOs the big picture, the things that you see as making you a great fit for their college.