As you may know it's up to the school to notify the local authority once you've asked the school to take your child's name off the school roll. The school isn't supposed to keep your child's name on the roll while the LA checks you out or anything. In theory the school could tell the LA that you are taking your child out "for negative reasons" or "because of poor attendance" but that makes no difference to your legal position, it just might mean that your first contact from the LA might be based on the wrong premise, which you should be able to counter by being unrelentingly positive and upbeat about the benefits and opportunities offered by home education! (Sorry v long sentence...need more tea...)
If you haven't already been given legal warnings about dealing with attendance (and no reason you should be from what you describe) then the LEA can't suddenly decide to make a fuss of it when you deregister her.
The law is different up here in Scotland, so this is not immediately relevant, but the guidelines up here say they are not allowed to take poor attendance into account when looking at home education. There is a recognition that poor attendance may be part of the decision to HE - whether that's because of health, bullying, school refusal or whatever - so they can't act like its suddenly a problem when you want to HE.
Do you have to give a term's notice? If so they might come after you for next term's fees. I would suggest a letter to them telling them they've failed totally to provide the level of education that you expect and as a result you're taking her out immediately.
Agree with reporting them . ISI and Ofsted won't really see turnover of pupils over time and school could come up with excuses anyway. They need you to make a complaint , preferably to governors first especially re. chaotic and inconsistent class organisation and poor teaching standards.
Since it's an independent school, the LEA won't have any involvement at all when it comes to attendance, it's just not their department.
The extra term's notice is a bloody cheek, but your contract with them is likely to be what you've said, a full term's notice, so it depends if you want to take the possible risk of legal action if you refuse to pay it. Personally I doubt they'd bother but they might if there are lots of parents in a similar position as they undoubtedly can't afford you all leaving and would not want to let you off with the payments.