With these kind of things, it's all about how you ask.
Writing in and demanding a refund due to illness and non-attendance =rude and not on.
Writing in to say thank you for running the trip, that DD was sorry to miss it and if there is money available for a refund/part refund after all costs are paid, you'd appreciate it, and appreciate them considering it, but understand if all of the £22 you paid was already committed regardless of whether she attended = fine.
Tone is everything in these communications. Showing appreciation for the effort put into running the trip and effort of them looking into your query makes all the difference. Too many people seem to have zero ability to phrase things in a polite or congenial manner, but come across as demanding and entitled.
So there won't be a clear-cut definite right answer to this question that applies to every trip in every school. That's why it's fine to enquire but not to demand or see a refund as a right.
Some trips are paid in advance and non-attendees have to be paid for as if they were there. Quite often a coach is booked and an extra being there or not won't affect the price the school pays. Quite often a school's ratios for trips is determined by expected numbers, so staffing (and therefore the cost of cover requirements) is determined by how many said they would go, not how many did go. That said, some locations bill after the event for the exact number of attendees, so part of the fee you paid might be refunded.
Do remember the £22 you paid probably covers child's entrance fee, so,ethi g towards travel, something towards adult entrance fees, something towards contingency and something towards subsidising those who don't pay. Probably the only part you can expect back is a small part for entrance fills if they weren't billed or a slightly larger part if more of the kids in the class paid the full fee than the school expected.
Personally, I'd write the money off. You were prepared to pay for her to go. It was a shame she will I'll admit missed out, but the money would have been gone anyway. Knowing how hard up schools are, I'd just let the money go as its not now an extra expense to yourself, just something you aren't receiving back, almost as a bonus. If it was a real push for you to pay that £22 then do ask and don't feel worried about doing so. Just expect that you probably won't get all of it, and try to be gracious and understanding if they can't give you any. It's not that they are setting out to take your money and they would return it to you if they could, but like most events we pay in advance for (cinema, tickets to advance to a theme park etc) if we are then a no-show for any reason, we simply expect to lose the money.