I had a child like that. It was really hard! Finding other children was the key to preserving my sanity. It required considerable investment of time and energy for a few years but has paid off in spades. Here are some of my techniques, and others that friends have used:
Hang around local children's play areas, especially at school/nursery chucking-out times.
Hire an older child to be a "trainee babysitter" by coming round to your house to play with your child after school while you get on with other things around the house. I paid them all a bit, but they said they would have done it for free. Many of them enjoy being looked up to and having a responsibility, and may value the chance to get some experience against the day when they will want to work as babysitters. If the age gap is big enough then in a few years you might feel comfortable having them take your child off to the park without you.
Tell all your friends that you are happy to look after their kids occasionally after school as a favour. Whatever the age of their children, having them round may interest your daughter. As a bonus, you get the parents' gratitude and they may return the favour by having your dd round sometime.
Tell everybody at the home ed groups that you are looking for friends for your daughter. Don't be too restrictive about age and gender. If you are willing and able to travel, say so. Other parents may be in the same boat but perhaps feeling too shy to say so or the subject hasn't come up, or they think you live too far away. Or they may be willing to put you in touch with acquaintances who've said their kids need more friends. At HE groups, be extra welcoming to anyone new to HE. (Not that you wouldn't be anyway!) People who are just starting HE or considering doing it are often quite worried that their kids may have no friends and will be eager to pursue any opportunities.
Go to some HE groups, museum visits etc even if they aren't quite your cup of tea, just to meet new people.
After-school childminder. I was working part-time from home and there was no way my dd was going to let me have any peace, so I sent her to childminders while I worked. She got her "fix" of other children and also enjoyed the childminders' attention.
I've forgotten what Lucy's school experience was like: would she be happy away from you in a large group without the academic emphasis? There are "after-school clubs", childcare usually based at school. Most of those attending do go to the school but the clubs are generally run quite independently and will accept any child. You could send her to one of those for a couple of hours once or twice a week.
One of my HE friends used to volunteer to man a local toy library. Her kids played with everyone who came along (and with all the toys, of course!) People usually visited for half an hour or so. It was mainly people with toddlers; does your dd like toddlers?
Finding friends for my daughter has been a life-saver. Not only did it get us through those difficult few years of repetitive role-playing, but it really paid off in future years. From the ages of seven to ten my highly sociable daughter spent a very large proportion of her time with her friends, not bothering me developing her independent socialisation skills. Now she is old enough to go places on her own it's even easier: she walks, cycles or takes the bus to friends' houses whenever the spirit moves her, or they meet up in the park. Some of their role play has been tremendously interesting to watch. They could act out a scenario for an entire day, sometimes picking up where they left off a week later. They launched into role play with just a few words, often adjusting storylines if they found themselves down a dead end. ("No, let's say I was really your sister that you hadn't seen for years because I'd been kidnapped...") As my dd became older the role-play transformed into creating plays and films.