Hello and welcome! I'm sorry to hear that your daughter is so unhappy at school. I think that when a child is miseraable at school, leaving that environment has got to be a good thing. She's unlikely to be learning much if all her mental energy is occupied in just coping with a difficult situation. In such a case, even if home ed isn't perfect, it must be better than the alternative.
It seems to me that how you approach home education will be affected by the question of how long you expect to be doing it. What's the waiting list like at the school you want to get her into? Is there much movement in your area? Is the school or the council able to give you any idea of how long it might take to get your daughter in there? Are there other schools you would consider if it looks like she'd be waiting a long time to get into that one?
If your daughter will probably be in school soon, then you may just want to keep up the basics of reading, writing and maths. You can get workbooks at WH Smiths and this might help you address anything your daughter needs to work on. Alongside that, you could take advantage of this opportunity to give your daughter some fun educational experiences which aren't so easy at school. For example, you could do some day trips to places of interest or visit local museums or art galleries. You could let her throw herself into some subject she really loves. For example, my daughter is interested in sea and pond creatures at the moment. We have been borrowing books from the library, watching videos, reading all about them on the internet, and going to aquariums. She has a pond and some goldfish, and is about to hatch out some Triops.
There is no legal definition of full-time education. The benefit of one-to-one teaching is huge, so much so that when ill children are being tutored by the LA, the LA is only required to provide a minimum of five hours per week of tutoring to them. Also, education doesn't have to look anything like school. Children are often learning when they aren't sitting doing formal work with an adult, and this counts as education also. In fact, some HE parents like me do no formal education at all, but instead let our kids learn by following their interests, asking questions, going places, reading, and playing. We don't see education as distinct from the rest of life, so it would be impossible to say how many hours per week the child is being educated. (How many hours per week do you spend "parenting" your child? Educating can be seen in a similar way.)
The LA has no legal duty to monitor home education on a routine basis. Most LAs contact parents soon after children are removed from school. One reason for this is to ensure parents haven't been pressurised into taking their kids out of school when they don't want to: schools have been known to do that. Another reason is safeguarding: the unfounded belief that home education is a somewhat dodgy thing to want to do and that parents might be doing it to hide abuse, forced marriage etc.
Your LA is likely to offer you a home visit to discuss your child's education. This is optional. If they TELL you they are visiting and don't make it clear you have a choice, this is an indication that they aren't inclined to follow government guidelines and you should be very wary of them. Some LAs behave really dreadfully because there is no comeback on them for doing so. On the plus side, if you take advice you can write assertive letters to get them to back off. When there is no legal basis for their demands, there is nothing they can do to you. You should never ignore a communication from them.
Some LAs say they want to visit your home to offer you "advice", but unfortunately they tend to be completely untrained and know little about home education, so their "advice" can do more harm than good. Most LA home ed staff come from a school background and don't understand home education very well. If you are after advice, you can get it online in a forum such as this one, or meet up with local home educators, who will be happy to chat with you. The best way to find them is to go on Facebook. In the search bar, type "home education" followed by the name of your nearest big town or your ocunty.
So long as the LA has not identified a specific concern about your child's education, they might get in touch once a year asking for an update to the information you provided. It's questionable whether there is a legal basis for this, but most home educators go along with it, often by providing a short written report. More often than annually is certainly out of line, unless of course they have raised a particular problem with the education you are providing.