OP, as it seems you may be home educating for a while longer anyway, perhaps it would be sensible also to look at ways you can improve your daughter's social life without school.
It's possible that in your area there is a sharp decline in home educated kids at secondary transition age. I don't think that is a typical pattern, however. Perhaps it just appears that way because you don't know the older kids? Some HE kids like a change and go off to school at secondary age, but by the same token some schooled kids who were happy at primary school find that secondary isn't right for them, and leave. My LA, like yours, has a significant HE population. Here, the LA's statistics indicate that nearly twice as many children are being HE in each year of secondary as primary!
Several years ago a well-connected long-time HE mum in my area asserted confidently that there were very few home educated teens around. But once people started asking questions, networking, and setting up the types of events likely to appeal to older kids, it became clear that she was completely mistaken. They appeared in their droves.
A friend of mine once observed that while you would expect kids to need less help connecting with their peers at secondary age, they actually seem to need more help. Adolescents become more particular about the company they keep and the activities they do. They become a bit more self-conscious. Yesterday I chucked my seven-year-old into the park with a handful of similar-aged kids who've just started home education, and they almost immediately began to play together. A few days earlier I had deposited her 14-year-old sister onto a sailing ship to spend a week with a dozen teens she'd never met. She's a very confident kid, but still she had worried: they all know each other, what if they don't like me, what if I say the wrong thing? As I drove off an hour later she was sitting nervously, not yet talking much.
What's more, some older kids have their own circles of friends and organise things for themselves. While they are perfectly friendly and willing to include any newcomers who happen to find them, they don't make much attempt to seek out and welcome new kids. You won't see their get-togethers posted to the HE list - they just invite their pals by Facebook. I am always nagging my dd to make more of an effort to accommodate new kids in the way that we adults do. It can take some perseverance to make a few friends and start learning about all that is happening.
My point is that perhaps there really are throngs of similar-aged kids in your area, but you might need to do some digging to find them. You could set up a group centred around your dd's interests and see if you can attract anyone that way: book club, maybe? Or encourage her to try some after-school clubs she might like. Take her to Hesfes if she'd enjoy that. I'm sure she can broaden her social circle considerably even without going to school.