Boarding schools cannot guarantee to turn the unmotivated or the lazy into the conscientious and hard working. It doesn't matter how much extra curricular stuff, inspirational teaching or non examined intellectually stimulating stuff they offer. Over the years I've seen quite a few who never pull their finger out and work despite everything the school and their parents do offer and try to do and change this. Some children are just inherently reluctant scholars.
Boarding schools (whatever their websites say) do not have a monopoly on teachers who are creative and engaging, they like day schools (in both sectors) will have teachers who are outstanding, good mediocre and poor. I also believe one mans inspiring creative good teacher may not be another ones.
Lots of boarding schools will offer a similiar programme like Trivium for yr 9. Basically they have to fill up the time, they've got your children 24/7 so there's bound to be more sport, extra curricular stuff, etc. Most schools don't want their pupils especially younger ones flopping around doing nothing. This is when they get home sick or get up to things they shouldn't.
Remember most boarding schools including nearly all you mention on here (bar 1 I suspect) are struggling to fill their vacancies, there is little appetite amongst U.K. parents for full boarding, very few have £37K+ (not including extras) that they can cheerfully stump up for the next five years, their websites are going to tell you how amazing their teachers are, many will have videos of earnest heads saying their school is totally unique, polished happy children also telling you that there school is totally unique, with vastly superior facilities to anywhere else, amazing teachers etc, all will tell you that their approache means that no child is left behind, that they will be reach their potential both in and out of the classroom etc etc. I actually think there's little to choose between most boarding schools, all you mention will have great facilities, good or excellent solid results reflecting the academic level of their intake, lots to extra curricular stuff, they'll also all have pupils who look you in the eye (there will be the odd one who doesn't but this is more likely to a about the child personality than the school) and I'd be surprised if all were not very welcoming (you're a walking £50 note).
It's the little things that will make it work for you and different from the next boarding school or your list. I personally think the HM is key, but you may have little say in who your HM is or you might spend hours deliberating over which one you like and she might leave during your DD's time there and be replaced by one you don't like. The children in your DD's house/Year are also important, but this is an uncontrollable factor. Try and find out how liberal or not they are (ideally they should be as liberal or not as you). How many rules there are in place or not that might irritate you and your DC. For example I know at least one of the schools mentioned insists it's pupils wear their uniform when they go to the local high street, (why?) that you have to go in pairs at least, I personally think this is totally unnecessary for 13+ others may definitely support this especially the idea of always being with other and in fact be uncomfortable if they weren't. DS's school did not allow children out at the weekend for special family events or let them finish term a day early for a flight home didn't bother me but I guess it could bother some. DS2's school was hands off parenting, suits me but not all, some would say communication with parents wasn't great, I work on the basis no news is good news, others want constant grade cards and updates a friend used to get "grade cards" every three weeks again totally unnecessary IMO. If you DD has a particular interest do check they offer it. I remember listening to a new parent at a school telling me their DS was an very enthusiastic basketball player, I pointed out the school don't offer basketball she looked very surprised and disappointed. Another friend sent her not very academic but talented golf mad DS to a well known school and then moaned that he could play golf as they didn't have a golf course! Even more bizarrely I've heard parents at DS2's school moan that their DS's can't be be day boys or flexi board, it's clearly stated on their website that it's a full boarding school you go through a complicated admissions process at every stage it's pretty bloody obvious it's a full boarding school. A parent once moaned on here that at DS2's school no softer options were offered in the 6th form, this is a school that's its well know that it's raison d'être is academia. Other parents moan about compulsory sport three times a week (not at DS2's school I hasten to add), this is standard stuff if your DD doesn't like standing on the hockey pitch literally come hell high water or freezing wind don't send them to a boarding school. You're stumping up a lot of money I'm stunned that people don't find this stuff out before they sign on the dotted line. It something matters to you ask. If you get this stuff right then hopefully your DD and you will be happy and thus become motivated in their studies. Don't go visiting schools and be blinded by Olympic sized swimming pools, rowing lakes or en suit bathrooms these are great but totally meaningless unless your DD is a obsessive swimming/rowing nut with serious personal hygiene problems.
As one parent once loudly and memorably said on a tour of a famous school "for £32k a year of course they're going to have manicured lawns, swimming pools and medieval books in their library what I want to know is what is the ethos underpinning this place?" Indeed.