I think I'd have a discussion with dd what she really wants to do.
It may be that she is trying all out for a term in order to cut back to a reasonable number when she's made an informed choice.
Also find out the costs-things like cookery might be a free afterschool one with just paying £5 a term for ingredients or something (have seen these).
I'd expect recorder to be at school and free probably-isn't it part of national curriculum they learn a bit in year 3(??), and Brownies is usually cheap-we paid £25 per term (may be slightly more the first term as they give a welcome pack). Gym is expensive though.
At 7yo my dc did a lot of clubs compared to others: Just thinking:
dd1: ballet, piano, tap/jazz, Brownies, panto (winter only)
dd2: ballet, trumpet, tap/jazz, Brownies, gym, panto (winter only)
ds: tap, Beavers, tennis, gym, panto (winter only)
Plus after school clubs (up to about 3 each-but those were free). So I'm guessing looking at the list that some of your dd's may be at school and free or very cheap.
What I would do is look at the list and discuss which your dd really wants to do (and isn't just thinking that would be okay as my friends are doing it) And choose to take payment for some of them. That means you pay the fees/get uniform etc. So if it was me I'd go for Brownies (uniform will be a big outlay at first, but they usually only have one), recorder, (with the thought of possibly changing it to an orchestral instrument if she gets on with it) and depending on price of recorder (I'd expect free/very cheap if it's school class learning, around £5 per 30 minutes if it's a group lesson/£11 per 20 minutes/£16 per 30 minutes if it's individual) and gym fees which will be the most expensive, but she will get a lot out of it, and if she gets better you may get to take her to competitions on your weekends.
That way you're paying for 3 of the new clubs, which sounds reasonable to me.
The advantage of taking on full payment of the lessons is that firstly, a lot of places like direct debit now, so it makes much more sense for you to put the money straight in than pay half and then your dd's dm has to pay it. I'd certainly be thankful for someone to remember some of the payments (but make sure you do remember at the right time!).
I think it also might make it nice for your dd to have a "daddy's special interest", but I may be shouted down on this.