What about setting the child <Grid>'s
HorizontalAlignment to Stretch, which is default, in fact?
Also you can always attach to parent <Grid>'s
SizeChanged Event[
^] and get value from
SizeChangedEventArgs[
^] or read parent
ActualWidth Property.
General Binding approach should look like this:
<grid width="{Binding ElementName=LayoutRoot, Path=ActualWidth}" />
but as I tested it, it worked only in VS designer and not in run-time. Why? -don't know :(