Hi there,
First, if it were me, I'd be sure to make sure there is nothing holding your daughter back inasmuch as any vision issues, etc.
Here's a question to consider: are you willing to follow through on your asking her if she wants to go to public school? I would keep that as an option, not as a punishment, but because she isn't letting you do your job, which is teaching she and her sister.
My guess is that this has become a habit each morning to whine and gripe. You will need to figure out how to not buy into the habit or the arguing. This might mean that you give her a chair to sit in, out of the way, until she's ready to come to the table and do her work. Maybe even hand her a note which reads: "You are welcome to sit here until you are ready to do your school without complaining. You may not play, you may not call out or interrupt us. Everyone else is wanting to learn. Come back when you are ready to learn too." Sometimes, holding back a favorite toy or activity until the work is completed is another option. For instance, our son only gets his Legos after his chores and homework are done each day. The Legos get put away each night before bedtime on school nights.
Another idea would be to curb her vocal complaining by giving her a book to write her complaints in. Enough self-indulgence and whining ("If you have a complaint, write it in your book; if you want to share it and find a solution to a problem, I'll be happy to look at it with you when schoolwork is done. If you just want to gripe and not show it to me, that's fine too. But you need to keep the negative attitude to yourself.") Sometimes, offering a reasonable outlet without being the sounding board ourselves neutralizes the back-and-forth arguments which rise from a situation like this one.
You might consider working on a behavior plan with her. Write: "When it's time for a task I don't want to do but must do, I can:" and then have her make a list of options for taking care of her feelings AND getting the job done. Have her show you the options and then make another list which features her ideas that you do like and approve of. Maybe she needs to run to the end of the street to 'get her mad out'. Maybe she needs to write in her complaining book for five minutes. Give her some authority over herself in helping her change her attitude.
Of course, there are the 'tough consequences' which sometimes come in handy. If she's wasting your time not doing the work, complaining, whining, make her trade you time. "We spent five minutes listening to you gripe, which means that you now may spend five minutes helping me clean the pantry" or whatever other task needs doing. When my son gets too self-indulgent and is using up my time with stalling, I have him 'pay' that time back to me. It might be less storytime that evening, it might be that he has to come and help me with housework. The point is less that they are doing something completely helpful, and more that we are inconveniencing them. A bit of conversation about the Golden Rule ("Oh, yes, I know you don't want to put silverware away right now, just like I didn't want to sit and listen to you complain for five minutes before your math got started. I guess if we treat others like we want to be treated, it works a lot better for both of us,huh?")
I wouldn't ask her to smile through it, I would ask, however, to be treated as she would like you to treat her. It might be funny for a day to go all Mrs Piggle-Wiggle on her and start emulating HER behavior-- just for a day, to show her how incredibly unpleasant it is. Stall on making breakfast, whine, complain that you HAVE to teach, "gawwwd! This is SO HARD and SO Boring! Why do you have to learn stuff anyway?" I'd employ this only if you want a playful parenting technique... don't 'go there' after she's already started stalling.
You might also throw out a carrot to her-- when she can do a whole week of school without complaining in the morning, there could be a small family reward, like going to a matinee on Saturday or taking her one of one for a cocoa outing and to the library to check out books, just the two of you. Good luck!