When I was in high school, my mom changed jobs. Since her work number was the primary "something is wrong with Dindrane" daytime contact at my school, somebody needed to update their records with her new phone number. I threw a little snit fit about my mom making me do it, because after all, I was not the one who had changed jobs and necessitated this bit of paperwork. I think I just refused to have it updated, but my mom also didn't have the school change it, and it went unnoticed because the school never had to call my parents during the day.
So, clearly, even when I was 15 or 16, and even when the thing I had to do was because of no action I had taken, my parents adopted the (entirely reasonable) attitude that I was perfectly capable of dealing with my school's administration, and it was therefore my job to do it.
Since I went to college in the same city that my parents live in (although I lived in the dorms rather than at home), they continued to insist that I be the one to deal with my school, even though they theoretically could have done some of the logistical things themselves. Even when it came to paying tuition (which they were paying for), I still had to be the one to actually deliver the payment. I didn't always like that they forced me to take care of things myself, but honestly, the only reason I ever had for not wanting to do it is that I am shy and it was hard and annoying.
It took until I for real moved away (about 2,000 miles away...) after college before it occurred to me that some things are annoying and hard no matter who does them, and it was never fair of me to expect my parents to do those types of things on my behalf once I was capable of taking care of them myself. I didn't learn that lesson for real until they were truly too far away to do more than give me advice, but it was a much easier lesson to learn because they'd been pushing me in that direction for so long.
So in other words, I think the only way to truly teach that concept to a child who does not understand it intuitively is to deliver some tough love, exactly like what you are doing. Not everyone will understand it intuitively, and that alone does not doom them for life, but those who can't figure it out on their own have to be taught.
I can also say that I completely sympathize with being in a situation where you are sick and there is nobody there to take care of you. It sucks. The first time I was sick in college, I had the biggest pity party you could possibly imagine. Even though my parents were in the same city, they didn't actually do anything for me (it was just a cold, and there wasn't really anything they could do). They did provide me with a lot of sympathy over the phone, and listened to me complain a little bit about how I had to take care of myself when I was feeling crummy. And then they told me to get some rest, get some orange juice from the dining hall, and go to the health center on campus for some medicine. They also advised me to maybe stock up on at least the medicine when I was feeling better so that I would have it on hand the next time I felt crummy.
So I think that providing sympathy for a situation that sucks (because it's no fun to be sick no matter the circumstances), providing advice on how to deal with it, and suggesting how to avoid at least some of the problem in the future is the best way to respond to this type of thing. If your daughter isn't interested in listening to any of that, then there's really nothing more you can say or do, and further discussion would be rather pointless. She won't want to hear you say that (I never wanted, and still don't want, to hear it when my mom says the exact same thing to me), but it makes you a very good mother if that's how you handle it.
Even now, I still feel the compulsion to state for the record that life is unfair, when I have to do something unpleasant. My mom is the most frequent recipient of that statement, but we've come to a tacit agreement that she will listen to me say it once, and I will stop whining once she has. And then I will take care of it (or not, and face the consequences).