I was ADHD inattentive and did not receive medication for it (story for another time, but I am passed that now). My wife is ADHD inattentive and she may be mild ASD, and she started her medication as an adult (post 30). Of my 3 daughters, the oldest is ADHD inattentive, another is ADHD inattentive and mild hyperactive (though some days it seems like anything but mild). My other daughter might be ADHD, but she focuses better than either of her sisters, so we are comfortable with her situation and are not worried about her progress in school.
To preface, I am not a psychiatrist, but given my circumstance I am fairly well versed in ADHD. I am not as familiar with ASD. My wife is on the spectrum, but it presents more as quirks.
For starters, the things that don't work: Reward systems. ADHD, especially inattentive types cannot control their inattentiveness. Taking away things based on their inattentiveness just causes them emotional pain and frustration because the inattention is not due to lack of discipline. They are often very aware that "their brains don't work right". Which was the exact way I phrased my own situation and how I felt about myself the entire time I was growing up.
Try to imagine yourself walking down the street minding your own business and someone comes up to you and gives you a cookie "as a reward". ADHD Inattentive types don't immediately make the association between the action and the reward. They are capable of drawing the line after the fact, but they didn't earn it through conscious behavior. In this example, you were walking politely and not yelling, which you often do. In this case, you were thinking of something serious and were just thinking about it quietly. ADHD Inattentive types often lack impulse control and their behavior is often simply a by product of their current emotion, be it happy or otherwise.
If your daughter is ASD and ADHD, she is likely in the gifted range and capable of comprehension of the medication you are asking her to take. In your case, you will have to find a way to reach her that works for her. If you don't want to be authoritarian (which I do not recommend anyway as it won't work for long term results), then you will have to find a different way to reach her.
This is where it gets tricky. As with ADHD, ASD also comes with certain situations where a non ASD person doesn't understand why they just don't get it. For example, in my wife's case, she works very hard to compensate for her lack of being able to pick up on subtle social cues that many of us take for granted. Your daughter may be presenting a strong will given that, to her, you are saying that 2+2 is equal to 5. Meaning, if she can't see the problem, your asking her to take medication is something that doesn't make sense to her.
In children, there are often little brain hacks that parents can do to promote the right behaviors and one is trying to see where the kids are coming from so that rather than using "because I said so" you can build on the child's current understanding to guide them into better behaviors, even if you aren't giving them the full picture.
For example, one of my twins, the ADHD both types one, was trying to do an art project and was getting stuck on a tricky glue / string part (she is 7). My wife, in her sweet but very thorough and logical manner, tried to explain all the different moving parts, but my daughter just grew more frustrated. My wife's answer was patient, compassionate and totally accurate at every turn, but she was using so much explanation that my daughter was simply not following any of it. I came in and gave her a short, almost wrong answer, but she took my 5-6 word explanation, asked me one question, which I answered, then after a small protest of me saying no I won't do it for her and repeating the short answer I gave earlier, she tried it and it clicked in her brain. After that, every thing my wife said before, my daughter did. Not because she remembered what my wife said, but the trick at the start was all she needed to have the rest fall into place.
This may not work in the same way with ASD because often times they require the long deliberate path. This is probably where your circumstance differs from my own. I don't get to dive in and help my wife in the same way I can with kids, but I know that what works for her does not work with my kids or myself. In this way, I would recommend working with a psychiatrist that specializes in ASD. To work with your daughter, you have to know the tricks that work with ASD.
Convincing her that education is paramount is your own fear being projected on to her and will never work because it is being told at her. Repetition and structure is likely the best way to get her on board. Using habits and scheduled repetitive behaviors can help with both ADHD and ASD. However, if you are like me, disorganized, you'll have your work cut out for you because it will be up to you to enforce the structure. Consult with your psychiatrist, but maybe showing her all the work she is doing will establish a pattern that will allow you to ask if you can "try medications to see if they can help change the pattern". I am disclaiming the hell out of that because my own experience with ADHD shows me that repetition can have negative emotional consequences if done incorrectly. The goal is to move her up and forward, not just point out where she fails.
Final message, and the TL;DR version: Work with an ASD psychiatrist and also maybe an ADHD psychiatrist, but also trust your instincts. Medication for things like ADHD and ASD is not an exact science. Every human is different. Different drug tolerances, different brain chemistry, behavior, health, etc. There is no one magic bullet. Some people get over it with eating and exercise (me, mostly, as I said, long story) while others go their whole lives (my good friend) and never really feel whole and end up with ups and downs (medication side effects) that they hate. For our family, we have chosen adderall (stimulant) based on wanting to avoid the non-stimulant and antidepressant medication side effects. My oldest stays up later than we want her to, but we feel like that was already an issue before we started medication, so we only give her a single morning time delay dose. My wife, who is on a higher, fast absorbing dose of the same medication, sleeps fine.
There will not be a one size fits all answer for your question. Take whatever you learn, and like a puzzle, see if what they say fits in your life and circumstance. If it doesn't, do some more digging. If at any point you aren't comfortable with your doctor, consider getting a second opinion if for nothing else than your own piece of mind. You already have your daughter's best interest at heart, so you are most of the way there.
