We are indeed, bhg. When I first signed up for library ebooks I was limited to 3 at a time for my choice of 1 to 21 days. Now most of the libraries offer 5 books at a time so I can download 30.
Excellent service.
And because my eBook reader isn't Kindle I can buy books from Waterstones, WHSmiths, Sony Reader Store, Foyles, Penguin ........and Amazon :)
Just an extra thought 2sp, if you register your new Kindle to your mum's account any books you buy will be available to both of you. If you prefer separate accounts there is a method of loaning from one Kindle to another but the book is only available on one Kindle at a time in that case.
Another thought - the Kindle Fire series can use books from public libraries BUT they are more expensive and have a much shorter battery life.