Oh no, how absolutely heartbreaking for both you and your sister (assuming of course your sister was quiltworthy and really cared about it in the first place). If this quilt were mine I would be furious with my dog if it had ruined such a beautiful quilt as this and she would immediatly be relegated back to crate training when I was not around to keep an eye on her and at night. And belive me I love my dog dearly but destroying such a beautiful quilt would definitely lose her all her free house roaming priviledges.
Is an entire corner missing? I can't think of any fix other than to cut off the bottom row and a half of blocks and bottom border where the entire corner looks to be missing. You could then unpick the quilting and use the fabric from the cut off borders that is still intact to remake the border and repair the bit at the top. You then would have portions of undamaged blocks that you could applique right over the damaged ones.
If you want to maintain the size, I am afraid any fix you do will look like repair patches, especially the entire missing corner, even if you could somewhat match the fabrics. You could probably find a nice compliment to applique something over the center areas but that missing corner where you had mitered the border print... There is not much you can do.