How do I databind a ColumnDefinition’s Width or RowDefinition’s Height?

Create a IValueConverter as follows:

public class GridLengthConverter : IValueConverter
{
    public object Convert(object value, Type targetType, object parameter, CultureInfo culture)
    {
        double val = (double)value;
        GridLength gridLength = new GridLength(val);

        return gridLength;
    }

    public object ConvertBack(object value, Type targetType, object parameter, CultureInfo culture)
    {
        GridLength val = (GridLength)value;

        return val.Value;
    }
}

You can then utilize the converter in your Binding:

<UserControl.Resources>
    <local:GridLengthConverter x:Key="gridLengthConverter" />
</UserControl.Resources>
...
<ColumnDefinition Width="{Binding Path=LeftPanelWidth, 
                                  Mode=TwoWay,
                                  Converter={StaticResource gridLengthConverter}}" />