If you keep the new device registered to your account, you don't have to go through emailing books to anywhere. When you purchase the book you can select
[Deliver to] Mom's Kindle and you're done with it. If it's a book you already purchased, you can go to books under Manage Your Kindle, find the desired book and under Actions, select
[Deliver to my:] and select Mom's Kindle. The books will be queued for delivery, and whenever she turns on the wifi, the books will be delivered.
I think that made sense... and makes sense, too.
I didn't think about it before, but... Did her original K3 have 3G? Will the replacement have 3G? If not, be prepared to walk her through configuring wifi. (yeah... I like to worry about everything.)