I have come rather late to this party but I will throw my two cents in.
In my experience, within my family (most of my generation were sent private), private schools are worth the money.
However, before you decide, do compare academic results between the schools in a variety of ways. Look at overall, look at value added, if possible look at people who have similar children to your DC and see how their kids turned out. Ask about pastoral care and bullying policies, ask about leavers destinations (beyond sixth form) and ask about disciplinary policies. Ask about homework loads, ask about learning support, ask everything of any school you're considering!
There are some crap private schools out there. Ask old girls of the school (recent ones) how their experience was.
If the school is a good fit for her, she will thrive (this is true of any school, of course) and their should be a good programme in place to deal with any bumps in the road, as well as good programme to help with her interests or gifts (it helps if the school allows children who aren't necessarily the best to be involved, e.g. a novice player in the chess team, a quiet girl in the debating team, a drama queen on the exchange, etc).
I can pm you the "why you should attend this school" page from my sister's old school, if you want, which gives you a general idea about private schools, but here are some other resources:
All American sorry, didn't realize this until now:
This Whole Website, Actually, Not Just This Page
here
here
Just remember, as these are from the US, public=state.
Personally, I think the big thing you get is a tight community feel with great school spirit. Everyone shows up for events, supports student initiatives, have the same set of ideals/morals and stick to them (e.g. studying is good, bing drinking is uncool), helps each other study, it's not always a walk in the park but you're all in it together, there's a tangible but indescribable feeling where you're at a school with a strong school spirit. You're not guaranteed to get this at a private school, nor are you guaranteed it to get it a state school, but I think chances are higher if you go to a private school to get this feeling. And it's worth every penny.
You said boarding is off the table, but how about for sixth form? You may both feel differently when she is approaching the end Year 11 and unless your finances chance, you will need to be planning ahead for a Year 12 boarding bursary.
Anyway, in your circumstances, I would wait to see if I got the bursary before setting your heart on a school (trust me, it's CRUSHING as a child if you get rejected due to finances on a school you've fallen in love with) and then decide from there. Ultimately, your daughter's happiness and education come before every other aspect.
(And don't be surprised if you get in and it isn't all wonderful from the start. At major transition points in my old school (year 1, 4, 7, 9) about half of new arrivals took over a term to catch up academically and settle socially. It can be tough but it's not unusual.)
Sorry for the long message, I got carried away as per usual.