I liked reading about three things when I was ten: horses, creepy dolls and time travel (in a personal rather than sci-fi setting).
Not sure if she likes horses (in which case she can read A Horse for Josie Moon), but I’d recommend A Handful of Time by (Canadian!) Kit Pearson and The Doll in the Garden.
My son, now 11, liked this book very much. He’s also into the Geronimo Stilton series, about a mouse reporter/crimefighter/adventurer. E.B. White’s The Trumpet of the Swan, Stuart Little and Charlotte’s Web might be a bit below your niece’s reading level now, but they’re wonderful books for most kids around that age. If it’s too soon for Tolkien’s engrossing but massive LOTR, start her off with The Hobbit.
Erin Hunter’s Warriors series, about a clan of feral cats, is also a favorite of my son’s, as is Charlie Higson’s Young James Bond series (you should start with Silverfin). I never read Richard Adams’s rabbit epic Watership Down, but my sisters loved it at that age. My niece, who was a little older than yours at the time and is a voracious reader, loved Garth Nix’s Sabriel.
My son loved Peter Pan and also the various sequels that have come out, Peter Pan in Scarlett and the books by Dave Barry and Ridley Pearson.
I loved that when I was a kid (I loved any book of Greek myths). A few years ago I saw D’Aulaire’s Book of Nordic Myths but didn’t get the chance to skim through it.
Pratchett’s “The Amazing Maurice and his Educated Rodents” 
They’ve already been mentioned, but I’ll also put my “Yes! These!” vote in for The Wee Free Men, A Hat Full of Sky, and Wintersmith.
According to Wikipedia, the complete series is:
* The Wolves of Willoughby Chase (1963)
* Black Hearts in Battersea (1964)
* Nightbirds on Nantucket (1966)
* The Whispering Mountain (1968)
* The Stolen Lake (1981)
* Dangerous Games, published in the UK as Limbo Lodge (1999)
* The Cuckoo Tree (1971)
* Dido and Pa (1986)
* Is Underground (British title: Is) (1992)
* Cold Shoulder Road (1995)
* Midwinter Nightingale (2003)
* The Witch of Clatteringshaws (2005)
I have only read about *half *of these, so I will be off to amazon right after you!
I call this a “sort of series”, by the way, because typically a minor character of one book will become the major character in the next book–they are not all about the girls in Wolves.
This is an excellent recommendation. My 10 year old devoured these books and had the opportunity to meet the writer for this series the day before his 10th birthday.
It was quite a thrill for him.
Another series, if you can call a series with only two books out so far a series, is
Operation Red Jericho and [Operation Typhoon Shore](Operation Typhoon Shore) which are set in the 20’s ( or maybe the 30’s.) and it is an action packed, Indiana Joneseque adventure in exotic locations ( china and that part of the world) and two british orphans ( whose parents are officially missing, so they are in the care of their stern uncle, a captain of a ship with a ragtag crew.) and dealing with mysterious scientific items that could change the world (!!!11111!!!), a man in a straw hat and a secret organization!!!111!!!
It is quite a ride.
The Extraordinary Adventures of Ordinary Basil is either a fun read to yourself adventure that isn’t too challenging or a superdoubleplusawesome book to read aloud bekaus zer iz a mad gherman scientist. ja.
My daughter (9) just finished reading all four by Rick Riordan loved them immensely. She is currently reading ‘Children of the Lamp’, a series by P. B. Kerr, and loving it.
Another author to recommend is Kenneth Oppel - Silverwing, Sunwing and Firewing is one trilogy, and Airborn and Skybreaker are another series.
This may seem like a bit much, but I’ve seen fires go through a book slower than my daughter, and it sounds like your niece is the same. Thank Og for the library, or I’d be broke. My mother once made a crack around my sister about the amount of money all of us spent on books, to which my sister replied ‘Would you rather we were buying drugs?’. Never heard another word about the book budget… Many thanks to all of you for the reckies, now I’m off to the online catalogue.
She might enjoy The Chronicles Of Vladmir Tod, Eighth Grade Bites and Ninth Grade Slays by Heather Brewer.
It’s a new series still being written about a half human half vampire struggling to find his way, as well as through the last year of middle school and on to high school. Lots of sarcasm and humor.
Does she like cats? If so, I’d recommend the Warriors series by Erin Hunter. There are three collections of six books each (I don’t think the third one is finished yet), a couple of sidebar novels, some “nonfiction” guide books about the cat clans and such, and a manga series.
I like kids’ books and cats, and I just started reading this series a couple of weeks ago. It’s got some cat-on-cat violence in it (fighting between clans and such) but you’re going to find some sort of fighting in most of the books that have already been recommended.
I just discovered a book called The Tale of Despereaux by Kate DeCamillo and think that it would be perfect.
I loved the Hobbitt when I was ten. I’d say start there for Tolkein. I tried wading through TLOTR, but only made it half-way through the trilogy, but the Hobbit was just right.
I also second the Tiffany Aching Books and Maurice and his Amazing Rodents by Terry Pratchett and the L’engel books.
I realise these are not SF&F, but can I recommend the Swallows & Amazons series by Arthur Ransome? They’re set in early 20th century England, and cover a group of children aged 8 to 12 who go sailing during their summer holidays and get up to all kinds of innocent hijinks. I think the series eventually hit 6 or 7 books about the core group of kids and then another two or three that focussed on some supporting characters from the originals.
I was (and am) likewise a huge SF nut, but I devoured these books when I was your daughter’s age, and still go back and reread them occasionally. They’re well-written, directed at kids but with the assumption that the readers are highly literate so there’s no sense of ‘dumbing-down’. And also completely devoid of any questionable content if that’s a concern.
Thank you THANK YOU thank you! Amazon and American Express thank you also.
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Okay, the order has been placed - “The Warriors” Boxed set for her birthday, and “The Enchanted Forest Chronicle” boxed set for Christmas (I went with what I could find in boxed sets because they make such nice gifts).
My 13 year old niece is getting books for her next birthday, too (I ordered them now for the free shipping):
“Podkayne of Mars”
“Ethan of Athos”
and
“Earth Abides” (never too young to start on my personal favourite, the post-apocalyptic fiction 
)
Thanks for so many good ideas. You guys rock.