There are a variety of possible factors that could be hindering her love of writing. Since I don't know your personal situation I will just throw out a few questions and ideas to think about.
First and foremost, I would ask myself if I am providing my child with ideas that she enjoys feasting upon. If a child's studies are filled with rich tales and fascinating ideas, then she will want to talk about them and write about them. It becomes as natural as eating and drinking for them. It really does! They literally begin to burst with excitement and have to let it out.
BUT, are you making it too academic? Children feed on ideas but hate the structured writing programs and systems that the educational arm of our society has set up in order to teach writing (and other subjects, for that part). Most great writers of the past never used these programs. They read great literature and were allowed to write about the subjects that truly interested them. This is not to say that formal instruction should never be given, but there is a season for it which I believe is when they are older and have already acquired the love of writing.
Are outside activities and entertainment not allowing time for her to be 'bored?' Children need to have enough alone time to propel them to make good use of their time. They eventually learn to pick up that neglected journal or other worthwhile pursuit that simply could not compete with the excitement that continually tempts them in today's world. I have always made sure my children are not always running here and there with friends, but have certain times during the week when they must create their own fun-- and technology is not allowed to interfere with this time. I know this is becoming very difficult in societies today, it is in your child's best interests if you make a stand against an 'entertainment-filled' life or they will miss out on its richest pursuits.
The writer needs to publish. She needs an audience. She may not have one immediately, but if she knows that eventually it will be read by others, she will spend hours perfecting the piece. Blogging is an excellent medium for this; writing contests, too. Even written narrations should be carefully preserved and those that she deems 'my favorites' should be read aloud to family members or sent via email, etc…
Children's wills are weak compared to their natural physical desires and emotions. As they go through life, the will is strengthened through training and discipline so that the emotions do not rule OR it is left weak and emotions and physical impulses are King. Unfortunately, our society is filled with grown adults like this today. Laziness and ease is a natural human tendency. The will must gradually be trained to enjoy mind exertion. Very few, if any, will love to learn the art of writing with absolute ease. But if good work habits are instilled while young, it paves the way for an enjoyable educational experience. Of course, even the strongest willed student can buckle under that ugly word, "academic," so we should be careful that this word is used as sprinkles on the cake of IDEAS, rather than the cake, itself. For then, it is no longer a cake, but a bowl of gruel.