It's hard to advise on livery costs, as they vary quite a bit depending on the area of the country you are in. However, for an 11yo/ young teen without experienced parents you would realistically need at least part livery (so there is someone on site able and willing to help if she gets into any difficulties) which may well cost £80+ a week. This may include bedding and hay, but if you want a specific type of bedding (e.g. dust free) this will usually cost extra.
Other regular costs:
Shoes: £60-80 every 6-8 weeks (some ponies won't need to be shod behind which will halve this cost, but most will need shoes in front if she's riding on the road).
Worming: This can vary a lot depending on how you/your yard decide to worm, but I would budget at least £20 per quarter for this.
Vaccinations: Say around £80 a year (£50 for vaccinations, plus vet call out fee. You may be able to reduce this slightly if you are able to share a call out with others on the yard)
Insurance against vet fees: £50 a month (this can be lower or a lot higher depending on the horse's medical history). Obviously this is not compulsory but vet fees can quickly grow into the thousands so it's a good idea for less experienced owners.
So as a rough estimate you are probably looking at about £350 a month just in regular costs, not including any tuition, entry fees or fun stuff. Keeping a horse can be done cheaper than this, but probably not by a teen with an inexperienced family.
On top of this you will have occasional/one off costs such as:
Horse purchase: I would budget at least £3500 for something safe and sane with a bit of experience in the things your daughter would want to do, plus transport to your yard and a vetting. You might be lucky and find something cheaper but cheap horses/ponies are often cheap for a reason.
Tack: Obviously some ponies/horses are sold with their tack but this will add to the purchase price- tack may also wear out and need replacing or mending. I'd budget at least £500 just for the basics.
Rugs: Again some ponies will come with a full wardrobe, but this is pretty rare, so I'd expect to spend at least £100-150 on rugs in the first year and again, these do break so may need fixing and occasionally replacing.
Other stuff: Depending on what the yard supplies and your daughter already has, I'd expect to spend at least £100 on miscellaneous "stuff" like feed buckets, grooming kit, yard tools etc etc.
£50 for 4 lessons a week and learning a lot about stable management is not a bad deal, and as your DD gets older she may well be able to exchange her labour for rides in other settings. When she is a bit older you could also look for a more formal share or part loan- maybe building up to a full loan when she is say 14.
Do bear in mind there are downsides to horse ownership- what if, for example, her pony goes lame and she has nothing to ride all summer? What if you buy something unsuitable that scares her and/or is unsafe? What happens if you, as a family, want to go on holiday? What if you have a sudden change in circumstances and can no longer afford the pony?
As a parent, I do think you also have to take some responsibility, learning about horse welfare, common medical problems and other issues (e.g. saddle fit)- if you are paying the bills, it will be you who gets asked, for example, "Should the horse have colic surgery or should it be put down?" and have to make that decision and justify it to your daughter.
Anyway, I hope this helps at least a bit. I was like your daughter and never had a pony of my own as a child. I still ride and have worked with horses and have been able to own as an adult when it fitted my lifestyle.