I was a child who went through divorce when I was 5 (now mid-20s) and exhibited many of the same types of issues. I went from being a straight-A student to skipping school and only passing because I picked up enough material just sitting in class to pass each test. Luckily this came late enough in high school where I had already been so scared at everything in life and so introverted that I was never in trouble and coasted all the way through 10th grade without trying, but eventually led to me being hospitalized when I skipped school to sit around at home thinking suicidal thoughts one morning.
I have to say that I think Crown and mikemorrell hit the nail on the head in each of their posts. They approached it a bit differently, but the end result remains basically that your child seems to be extremely bright, is probably not being challenged, and is most likely locking herself away inside her head and lashing out at the world because of the sense of abandonment.
I can't honestly give advice because I don't have kids, and what my parents did for me didn't really work. It wasn't until the past few years where I actually built my confidence and started figuring out who I am, especially in the past year or two (and ESPECIALLY since I found out that there were actually other people like me). So, I can only say what I think might have worked or what certainly doesn't work.
I can tell you that punishment does not work and will only further the problem. And while it may be hard to understand, INFPs look at everything in the world much differently. A perfect example that mike brought up was how your daughter said "I already know the subject matter, why should I have to do these dumb assignments?" and he similarly thought there was a lot of merit to that (I think that too). Throughout my entire life, even up until now, actually being forced to go through with things that make no sense to me has been beyond aggravating. I feel like my time is being wasted and I'm being treated like I'm stupid, even if only indirectly. I remember going through school looking around my class and wondering how people just didn't "get it" like I did. And then, I realized that the reason I was forced to do all this was directly because of everyone else being "stupid," so I privately gained an ego within myself and became annoyed at everyone else. To this day I fight with this side of myself. I am constantly going through life just wanting to shout "Are you serious?! I mean, DUH!" at pretty much everyone.
Of course now I know better and that just because someone isn't as smart or talented at things I am, that doesn't make them any less worthy of a person and we are all equal although different. But when you are 11 it is a bit harder to figure that out. You simply think "No one understands me, no one really cares because they if they did they wouldn't do these things to me (like punishment)." Part of the acting out being exhibited is probably just genuinely your daughter growing up and testing you, and there is always going to be that, but also she is figuring out who she is at that age, and as she does she is most likely realizing the world seems like it doesn't agree with her and how she thinks at every turn.
So, punishing your daughter for doing things that seem completely illogical to her really just reinforces that you're just another idiot to her, thus adding to the reasons not to actually listen to you. You are going to have to find a way to address two major things. First, that you are always on her side and have her back, even if you just go "Yes, I agree with you, but even though the system is stupid we still have to do it because -reason-." But you have to be genuine that you mean that and aren't just saying it to get her to shut up and do something, because INFPs can see right through that. Second, you need those reasons to make sense and show another side to the task being requested, or find ways to help her cope with doing mundane things. As was pointed out at the end of mike's post, the INFP has to buy in freely and have it agree with one of their principles somehow.
You also may need to get used to a few things that might be hard. I know ENFJs usually are pretty big on housekeeping and organization, but you may have to meet half way and allow the room to be a little messy and the bed to not be made sometimes. I can only speak for myself personally, but one of the reasons I signed up to this board a few days ago was to eventually write a post asking for help with how other INFPs have been successful at organizing their lives. Organization is just another word for structure to me, and I HATE structure.
I know this might sound like a weird suggestion, but I have to say that recently I have been extremely intrigued with how the French live. Just hear me out if you are of the "America rules!!!!" crowd before rolling your eyes and going to the next post. I had noticed a show on the Cooking channel (I love to cook!) called "French Food at Home," where the host, Laura Calder, was so authentically happy and pure that I was immediately drawn into the show and immediately looked up everything I could about her. She was from Canada, but went to visit France for a few weeks and wound up staying 7 years prior to doing her show.
There is a term often attributed to the French which I'm sure everyone knows called "joie de vivre," or joy of living, and so many people talking about her show on the internet kept commenting about how she had it. Part of the joie de vivre of the French is taking interest and gaining happiness in the small and mundane details of everyday life. As INFPs are big on perfection, I have found that by wanting things to be perfect for myself (as opposed to, say, because other people said so or it is what the norm is), I find myself doing more maintenance and keeping my house tidier. I noticed that I started cooking and trying to emulate Laura's happiness and acted how she did, with a certain goofiness almost, and lo and behold cooking (and by extension, life) has become even more fun for me. I started making my bed recently because I enjoy making it perfect and then it is still perfectly waiting for me when I'm ready for bed. A brief thing I had read in a review for a book I intend to purchase mentioned how the author discussed putting lavender scent on your pillow. This actually turned out to be a selling point to me. As a guy, I'm not sure how much I care to put lavender scent on my own pillow, but I understood the purpose behind it was doing something so small and effortless while basically pampering yourself. You could probably play on her "princess" desires by doing something similar that allows her to be a little bit of a perfectionist and spoil herself at the same time. Perhaps you can even divvy up the chores and give her more that are in her best interest, that play on the INFP perfectionist tendencies, or that at least make sense to her.
It may also help to try to envision her as a 13 or 14 year old when deciding how to treat her. While the actions she is displaying certainly may seem immature, the mindset and thoughts behind those actions are probably much more advanced and complex. Skipping school to the outside world might seem immature or childish, yet through the eyes of an INFP it might be completely illogical to go to school in the first place, so "how stupid is everyone else to do it just because that's what people say they should do?"
Lastly, encourage creative activities. If she isn't already, find out the most artistic medium that SHE wants (and it HAS to be what she wants to do the most no matter how much you disapprove) and get her going. Whether that is playing an instrument, taking dance lessons, or painting or drawing. Losing myself in art, especially when I'm creating it somehow, is extremely cathartic, and this seems to be a trend among INFPs. A creative outlet is most likely essential to help her get her feelings out rather than bottling them up. Just remember, do not critique unless it is very helpful and warm criticism! Almost like just a suggestion, like "I really liked how you did it that way. Had you thought about doing it this way?" Otherwise, she will take it personally and be afraid to share her art with you from then on out, because for us it is an expression of our emotions and feelings, and it becomes a personal attack on our emotions. Or rather I should say it at least does for me, other INFPs may have thicker skin
I have to say I genuinely feel terrible for your little girl. I can't say I know exactly what she feels like, but if it was anything like what I went through doing the same thing, then it is very, very lonely. I would echo the sentiments about her eventually hurting herself if you let this go on or don't find remedies to the situation, and cannot understate the urgency with which you should act. This is certainly not to say she is going to do something tomorrow or even two years from now, but people don't get suicidal over 1 day just like obese people didn't get there in 1 day either.
I'm sorry this was so long, I can't help but write this way. But I sincerely hope this contributed in some way and hope you find harmony with your daughter
