Amen! I was coming in here to suggest that. They’re really delightful, geniunely funny and sympathetic books. The Wonderworks movie of it is also a great one (it runs during PBS fundraisers, and you can rent it at Blockbuster, etc.). Emily of New Moon is also wonderful (though a tad darker).
I should learn to read all of a thread before I go searching for something. “The Railway Children” really clivked with me, but, of course, I also was remembering “The Boxcar Children” instead. I go off seqarching for it for 30 minutes and someone mentions it a but further down the page.
But thanks for the memories!
I would second the James Herriot books, but he didn’t sugar coat the fact that there was quite a bit of death in his line of work. The books don’t dwell on death at all, but it is there and not all stories have a happy ending.
I’ll second Madeleine L’Engle, and note that in A Wind in the Door the boy (Charles) is sick and is saved by his sister (with the help of a cherubim).
My favorite of hers is A House Like a Lotus - the protaganist is a 16- or 17- year-old girl; it’s a story of forgiveness and acceptance.
I re-read my Little House on the Prairie books a couple of years ago. Still good reading.
Once again popping in to second someone else’s suggestion. All Creatures Great and Small is a perfect place to start on James Herriot’s books, then move on through All Things Bright and Beautiful, All Things Wise and Wonderful and The Lord God Made Them All. Each chapter in them is like a short-story unto itself; often hilarious and very touching. And true, which makes it even better.
What Katy did
Goodnight Mister Tom
Watership Down
I don’t know if she’d like The Prophet by Gibran, but my sister read it quite often when she was ill. I thought it was good, but not really something I enjoyed immensely.
And of course, I have to mention Le Petit Prince by Antoine de St. Exupery, translated as The Little Prince (though if your daughter happens to speak French, I recommend the original over the translation). It isn’t really a sappy getting-better story, and it is in fact very sad, but also very inspiring. and thought-provoking, if you take the time to think about the words. I’ve read the book several dozen times, and always get something new out of it.
I must have the same temperment as your daughter, when I’m feeling poorly, I need my comforting and uplifting books.
A few thoughts:
Then There Were Five by Elizabeth Enright. An orphaned boy is welcomed into a new family. This is the third in a series of books about the Melendy family, although each one does pretty well as a stand-alone.
Karen by Marie Killilea. This is the true story of Marie Killalea’s daughter Karen, who suffers from cerebral palsy but goes on to overcome the obstacles in her life. This is a little dated (poor Marie cheerfully smokes constantly) but it’s a very sweet story about family life. Faith is a big part of this book.
Mustang: Wild Spirit of the West by Marguerite Henry. Story about a young girl, recoving from polio, and her love of wildlife in general and wild mustang ponies in particular. This is especially good for kids who love ponies and horses, however I will caution that it includes a brief but scary scene of horses being ill-treated, prior to said horses being rescued.
I second Marguerite Henry with her book King of the Wind.
Hmm, I can’t think of many more in addition to all the great suggestions already here. Thinking of what I read as a child…
How about “Where the Red Fern Grows” by Wilson Rawls
Anne of Green Gables series by Lucy Maud Montgomery
or
The Black Stallion series by Walter Farley?
SkipMagic, King of the Wind was absolutely my favorite book when I was a kid. I read it over and over again. 
Classics along the lines of getting better through God’s grace (or there abouts) there’s What Katie Did by Susan Coolidge http://www.amazon.com/exec/obidos/ASIN/0140366970/qid=1051311322/sr=2-1/ref=sr_2_1/002-7987270-7572803
And The Little Lame Prince by Miss Mulock (I loved this story as a child, and reread and enjoyed it still just a few months ago)http://www.amazon.com/exec/obidos/ASIN/1589632907/qid=1051311622/sr=2-1/ref=sr_2_1/002-7987270-7572803
Then there’s dealing with the aftermath of an accident Johnny Tremain by Esther Forbes,
A much more recent, and very nice, book about getting over illness is Fever 1793 by Laurie Anderson http://www.amazon.com/exec/obidos/ASIN/0689848919/qid=1051311899/sr=2-1/ref=sr_2_1/002-7987270-7572803
This last book- one more of a good vs evil theme than illness- which is a book that many people who don’t like the horror novels still like quite a lot (oddly, I think it’s better received by kiddos and non-fans than his normal audience) ** The Eyes of the Dragon** by Stephen King. After I finish the 50(new) book challenge, I think I’ll be reading this one again since it’s been a few years 
http://www.amazon.com/exec/obidos/ASIN/0451166582/qid=1051312573/sr=2-1/ref=sr_2_1/002-7987270-7572803
I’ll third Marguerite Henry with any of her books, really - I read and enjoyed them all as a child…but I was and am a horse nut, so there ya go.
I’m also going to toss out The Witch of Blackbird Pond by Elizabeth Speare.
On Marguerite Henry, you also want Misty of Chincoteague and its sequels, Sea Star: Orphan of Chincoteague and Stormy: Misty’s Foal. And especially Cinnabar: the One O’Clock Fox, which isn’t about horses but is just great fun.
Ooh, now that I see chicque’s post, I have to mention The Sign of the Beaver which I think is also by Elizabeth Speare.
Has anyone else mentioned Little Women? If so, I didn’t catch it. I’ve read that book so many times… Beth, one of the sisters in the family, is chronically weak (if not actually ill. Her sickness is a little mysterious. I think I always assumed she either had migraines or TB). She does pass away in the course of the book, but it’s an excellent read. Jo was always my favorite sister – is that true for everyone? Or does it vary by reader which sister is the most interesting?
I was never satisfied with whom Jo chose as a husband. It made sense, but still.
Mrs. Furthur
I’ve been thinking of your daughter all day and thought of two other delightful, sweet books with children and animals: My Family and Other Animals by Gerald Malcolm Durrell. It’s nonfiction, about his family’s stay on the Greek isle of Corfu in the 1920s(?) and his encounters with the various interesting people and animals that lived there. Side benefit: you learn a lot of insect and animal classification!
I’m pretty sure I read two books, but looking now on Amazon, I’m not sure what the other one might have been. A likely suspect is Birds, Beasts and Relatives but the title isn’t ringing a bell. Here’s the whole page if anyone would like to take a look: Books by Gerald Malcolm Durrell
One of my favorite books in the world is Addie Pray by Joe David Brown. It’s the novel the movie Paper Moon was based on, so as you can imagine, its main theme is swindling people out of their money in the Depression-era South. (The movie ends halfway thru the novel and doesn’t touch the main plot line that leads to the climax.) Despite its “criminal” hero and heroine, the book has a sweetness and a charming, happy ending. I love reading it every few months for a pick-me-up. It’s a definite comfort novel for me!
Oh, right, I was thinking of the Boxcar Children. ONe of those nice stories where kids have adventures in the adult world but still pretty much live in the kid world.
I would like to recommend “The Little Locksmith” by Katharin Butler Hathaway and the “Maida” books by Inez Haynes Irwin.
The “Locksmith” book is Hathaway’s true story of being stricken by tuberculosis at age 5 just before the turn of the century and her eventual spiritual triumph over the disease.
I hope your daughter enjoys all these suggestions.
How about The Princess Bride by William Goldman or Bridge of Birds by Barry Hughart?