Wow! I could have written your post! I, too, have an 11 year old who reads at least 10 library books a week and will lose this awesome resource when we start our open ended around the world tour in a few months. She doesn't have much knowledge about e-readers, but I looked into getting her one anyways, for this very reason. The whole library book thing has proved to be fruitless for me too. No libraries that I have access to have any children's books and the ones that do seem to have many books unavailable (maybe already signed out by other patrons?) In my internet search, I have found a few, perhaps, shoddy options where you can get many, many free books, including Harry Potter (which is what causes me to suggest that it is less than legit). I also know that you can download PDFs for free and I have been able to find some interesting articles for her- interesting as a different genre from fiction novels.
I did end up getting her a kindle for her birthday (this Friday- I'll tell you how it goes ;) ) Both my parents, myself and another friend are buying her gift cards to use towards buying books and I am sure a few $5.99 books will be bought. I cringe a little. I did buy the 3G version that you might want to consider too for traveling. If you haven't seen it yet, it offers free 3G wireless connection in a whole host of countries. The browser is very slow and not something one would want to use too much, but it does have things like wikipedia that my daughter likes to go to, and it can offer an option if you want to write a quick email while on the road.
Anyways, I hope that as the technology progresses, the prices will come down. (they didn't even really have any sort of "Black Friday" deal on books in the Kindle Store, urg) For now, I've added books as an expense line on our travel budget.