One hand alone music for beginning piano students is pretty hard to find. See the letters below for some suggestions.
I have been there. And I do not know of any actual METHOD book for a child with only one hand. The strange thing is that many composers have actually written rather DIFFICULT music for one hand (almost always the left hand, that I have seen), even classical composers. Visit this page on www.imslip.org and you will see a long list of piano works by classical composers just for the left hand.
The problem is that these are not beginner-level pieces! I am afraid you are going to have to make do with existing method books and adapt them for your student's use. What you are doing for her now, already, is surely the best thing for her.
Some skills will come sooner
Since she is able to focus on just one hand, your student will probably be able to learn some piano skills a lot sooner than a two-handed child. I'm speaking in particular of finger substitution (changing from finger 5 to finger 4, for example), and of crossing under and over, as for a full scale or a chromatic scale.
"On the Other Hand" by John Robert Poe really does have songs playable by just one hand. And they look intriguing; if you go to this page, you can scroll down to "John Robert Poe" and click the preview buttons to see the music sheets for one hand. The music looks pretty fun! Much of it students could learn by rote.
Look at the right column of this page
All the featured pieces and books in the right column are for one-handed playing. And a nice feature of the Sheetmusicplus website is that if you look at a particular book or piece, just like Amazon, they will show you similar items that were purchased by other visitors to that product.
What IS Five Finger Piano Music?
I have found that "Five Finger Piano music" doesn't necessarily mean any such thing - what it really seems to mean is five fingers times two = ten! But there is also "music for one hand", and if the fingering is changed, it could be for either hand. I suggest that you hunt for lead sheets and check the suitability of each song; I have a page which links to all my lead sheets.
Think DUETS for your one-handed student
Duets are obviously going to be a big part of her music. There's a chance she will play chords very early and love them. Have you looked at the Secondo part of my Greensleeves arrangement? It is an easy way to introduce broken chords, and could hardly be easier (except for the beautiful black note in the middle of the E major chord).
And PEDALLING for beautiful sonorities
Is she using pedal yet? Probably she can't even reach the darn thing - but it will make her one-hand-alone melodies and chords feel full and satisfying when she occasionally makes use of it, even if she has to stand up.
Start your search with these books
Check out some of the books and sheets at Sheetmusicplus that I have linked to on this page. Most of them are for beginning Elementary to Intermediate.
If you investigate "Left Hand Solos - Classical Themes", you may wonder a bit at the names of these classical pieces you have never heard of before! But despite their funny names, the songs are indeed classical. For example, "Plus & Minus" is by Schumann; "Out of the Past" is by Czerny, and "Memories of You" by Gurlitt.
Here are the featured songs in the John Thompson book "For Left Hand Alone", three of which you can look at in preview:
The Beautiful Blue Danube by Johann Strauss (This looks very nice, and not too terribly hard)
On Parade (looks very fun to play!)
Midnight Boogie by Jack Foy
Echoes From Schubert (based on the Erlkonig - it looks like a very full-sounding arrangement of that piece using scales and chord fragments)
The Ballet Dancer
On The Banjo
Farewell To The Piano by Ludwig Van Beethoven
In The Hall Of The Mountain King by Edvard Grieg
Other places to look
Another idea is to hunt in violin books, or the method books of other one-note-at-a-time instruments, for catchy melodies, and use them for the beauty of the melody or for improving notereading. Of course, you would have to write it out.
Even regular method books can be adapted... and have you looked at The Perfect Start for Notereading? Melodies are shared between the hands on most songs, but many could be played with one hand alone.
Take a look at the intermediate-level blues piece, Midnight Blues by Carol Matz: it looks like a lot of fun!
So what is possible for a one-handed pianist?
Watch the video of Nicholas McCarthy, below, a one-handed pianist who graduated from the Royal College of Music. An article from The Telegraph tells his story... He shows what is possible for a musician who plays one hand alone, if the desire is there.
A special one-hand-alone student such as yours challenges the imagination and your determination as a teacher... I hope these ideas are helpful.
Please note that all comments are moderated, and will not appear until I have approved them. Also, IF YOU ARE ASKING FOR MUSIC THAT IS NOT IN THE PUBLIC DOMAIN, YOUR REQUEST WILL BE IGNORED. That's pretty much any music written in the last 75 years...