No. You definitely don't have to give them all that information.
The children's school and families bill, which would have given LAs lots lots lots more powers over us all, just ran out of time in parliament and did not go through. There are many LAs who are flexing their muscles now, trying it on, acting beyond the law which came so close to being changed. That is what yours are doing.
You need to write back to them and say:
"thank you for your letter. We are providing XX with an education suitable to her age, ability and aptitude (plus any SEN if she has any). I enclose some information about this"
You then choose what you want to tell them, bearing in mind that this is what they will use to decide "oh yes, there's an education going on there" or "oh no, we must hassle this family and investigate further".
This page here will really help you decide what information you want to give them. Personally, if I got a letter from the LA, I would tell them about my educational philosophy (autonomous) and explain what I think that means. Because I think the privacy of my children is very important, I would not be telling some numpty who I have never met how my children spend their days, what they are learning at the moment, etc. But I would give them some examples of ways in which, say, a trip out could be educational, and give them an idea of how often we go on such trips, or tell them a little about how purposive conversation can be highly effective and efficient. I would tell them that social contact is very important to us as a family, and that we have a network of friends of all ages who we see very regularly.
I definitely would not be giving them anything about a proposed programme, methods used, or patterns of work. I might well be inclined to explain to them that explicitly theorised pedagogical methods, pre-planned timetables or work patterns, and curricula are much more relevant to school-based or school-style learning, particularly when one is trying to educate 30 children of different abilities at the same time. None of those things are as effective or efficient as responding to the interests and needs of my children in the moment. Of course I have materials on hand for when the moment of inspiration strikes, and of course we go out and get materials we need when we are forward planning for some activity, but I absolutely do not plan ahead.
Oooh, I think I'll bookmark this post for if/when the LA discover us...