Book Sharing

You mean she is tapping her icon at the top of the 'Reading Now' tab and your id ? Has she tried logging out and back into the account in the app ? If she has, what is the full text of the message she gets when trying to download a book you purchased ?

With Family Share the organiser agrees to pay for all of the members purchases (though if iTunes gift cards are available in your country the members could potentially buy them and redeem them onto their own accounts ; they cannot use their own debit or credit card) - you cannot just share past purchases.

She is selecting you name in the Purchased section in the Books app ?