Your template for the Slider has a pretty big lack of behavior. It draws itself, but has no triggers to define reactions to user clicks / drags, etc.
If you want to customize the look of the slider, I would suggest that you start with the original template and customize it. To do this, download the app Show Me The Templates!, extract the template to your project, then start customizing the thumb.
I hope this was helpful.
Edit: This is wrong. This only works in WPF. In Silverlight, the tool I'd recommend is Blend OR you can use Visual Studio's "Extract to Resource" feature on the template:
http://blogs.msdn.com/b/wpfsldesigner/archive/2010/06/03/creating-and-consuming-resource-dictionaries-in-wpf-and-silverlight.aspx
Edit Edit: I modified the builtin slider template to include your thumb style as a reference to the type of system of small tweaks you should go for rather than a "rip and replace".
<Grid>
<Grid.Resources>
<Style x:Key="ThumbStyle1" TargetType="Thumb">
<Setter Property="Template">
<Setter.Value>
<ControlTemplate TargetType="Thumb">
<Grid >
<Path Data="M13.832941,48.001114 L-0.00066526519,15.584446 L-0.1685528,3.1627214 L46.375076,3.1882343 L46.59383,15.594595 L33.375225,48.063122 z" Fill="#FF6D6D74" Margin="-0.668,2.662,0.906,-0.562" Stretch="Fill" Stroke="Black" UseLayoutRounding="False"/>
</Grid>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>
<Style x:Key="SliderStyle1" TargetType="Slider">
<Setter Property="BorderThickness" Value="1"/>
<Setter Property="Maximum" Value="10"/>
<Setter Property="Minimum" Value="0"/>
<Setter Property="Value" Value="0"/>
<Setter Property="BorderBrush">
<Setter.Value>
<LinearGradientBrush EndPoint="0.5,1" StartPoint="0.5,0">
<GradientStop Color="#FFA3AEB9" Offset="0"/>
<GradientStop Color="#FF8399A9" Offset="0.375"/>
<GradientStop Color="#FF718597" Offset="0.375"/>
<GradientStop Color="#FF617584" Offset="1"/>
</LinearGradientBrush>
</Setter.Value>
</Setter>
<Setter Property="IsTabStop" Value="False"/>
<Setter Property="Template">
<Setter.Value>
<ControlTemplate TargetType="Slider">
<Grid x:Name="Root">
<Grid.Resources>
<ControlTemplate x:Key="RepeatButtonTemplate">
<Grid x:Name="Root" Background="Transparent" Opacity="0"/>
</ControlTemplate>
</Grid.Resources>
<VisualStateManager.VisualStateGroups>
<VisualStateGroup x:Name="CommonStates">
<VisualState x:Name="Normal"/>
<VisualState x:Name="MouseOver"/>
<VisualState x:Name="Disabled">
<Storyboard>
<ObjectAnimationUsingKeyFrames Duration="0" Storyboard.TargetProperty="Visibility" Storyboard.TargetName="HorizontalTrackRectangleDisabledOverlay">
<DiscreteObjectKeyFrame KeyTime="0">
<DiscreteObjectKeyFrame.Value>
<Visibility>Visible</Visibility>
</DiscreteObjectKeyFrame.Value>
</DiscreteObjectKeyFrame>
</ObjectAnimationUsingKeyFrames>
<ObjectAnimationUsingKeyFrames Duration="0" Storyboard.TargetProperty="Visibility" Storyboard.TargetName="ThumbDisabledOverlay">
<DiscreteObjectKeyFrame KeyTime="0">
<DiscreteObjectKeyFrame.Value>
<Visibility>Visible</Visibility>
</DiscreteObjectKeyFrame.Value>
</DiscreteObjectKeyFrame>
</ObjectAnimationUsingKeyFrames>
<ObjectAnimationUsingKeyFrames Duration="0" Storyboard.TargetProperty="Visibility" Storyboard.TargetName="VerticalTrackRectangleDisabledOverlay">
<DiscreteObjectKeyFrame KeyTime="0">
<DiscreteObjectKeyFrame.Value>
<Visibility>Visible</Visibility>
</DiscreteObjectKeyFrame.Value>
</DiscreteObjectKeyFrame>
</ObjectAnimationUsingKeyFrames>
</Storyboard>
</VisualState>
</VisualStateGroup>
<VisualStateGroup x:Name="FocusStates">
<VisualState x:Name="Focused"/>
<VisualState x:Name="Unfocused"/>
</VisualStateGroup>
</VisualStateManager.VisualStateGroups>
<Grid x:Name="HorizontalTemplate" Background="{TemplateBinding Background}">
<Grid.ColumnDefinitions>
<ColumnDefinition Width="Auto"/>
<ColumnDefinition Width="Auto"/>
<ColumnDefinition Width="*"/>
</Grid.ColumnDefinitions>
<Rectangle x:Name="TrackRectangle" Grid.ColumnSpan="3" Grid.Column="0" Fill="#FFE6EFF7" Height="3" Margin="5,0,5,0" RadiusY="1" RadiusX="1" Stroke="#FFA3AEB9" StrokeThickness="{TemplateBinding BorderThickness}"/>
<Rectangle x:Name="HorizontalTrackRectangleDisabledOverlay" Grid.ColumnSpan="3" Grid.Column="0" Fill="White" Height="3" Margin="5,0,5,0" Opacity=".55" RadiusY="1" RadiusX="1" Visibility="Collapsed"/>
<RepeatButton x:Name="HorizontalTrackLargeChangeDecreaseRepeatButton" Grid.Column="0" IsTabStop="False" Template="{StaticResource RepeatButtonTemplate}"/>
<Thumb x:Name="HorizontalThumb" Grid.Column="1" Height="18" IsTabStop="True" Width="11"/>
<Rectangle x:Name="ThumbDisabledOverlay" Grid.Column="1" Fill="White" Opacity=".55" RadiusY="2" RadiusX="2" Visibility="Collapsed" Width="11"/>
<RepeatButton x:Name="HorizontalTrackLargeChangeIncreaseRepeatButton" Grid.Column="2" IsTabStop="False" Template="{StaticResource RepeatButtonTemplate}"/>
</Grid>
<Grid x:Name="VerticalTemplate" Background="{TemplateBinding Background}" Visibility="Collapsed">
<Grid.RowDefinitions>
<RowDefinition Height="*"/>
<RowDefinition Height="Auto"/>
<RowDefinition Height="Auto"/>
</Grid.RowDefinitions>
<Rectangle Fill="#FFE6EFF7" Margin="0,5,0,5" RadiusY="1" Grid.Row="0" RadiusX="1" Grid.RowSpan="3" Stroke="#FFA3AEB9" StrokeThickness="{TemplateBinding BorderThickness}" Width="3"/>
<Rectangle x:Name="VerticalTrackRectangleDisabledOverlay" Fill="White" Margin="0,5,0,5" Opacity=".55" RadiusY="1" Grid.Row="0" RadiusX="1" Grid.RowSpan="3" Visibility="Collapsed" Width="3"/>
<RepeatButton x:Name="VerticalTrackLargeChangeDecreaseRepeatButton" IsTabStop="False" Grid.Row="2" Template="{StaticResource RepeatButtonTemplate}"/>
<Thumb x:Name="VerticalThumb" Height="11" IsTabStop="True" Grid.Row="1" Width="18" Style="{StaticResource ThumbStyle1}"/>
<RepeatButton x:Name="VerticalTrackLargeChangeIncreaseRepeatButton" IsTabStop="False" Grid.Row="0" Template="{StaticResource RepeatButtonTemplate}"/>
</Grid>
</Grid>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>
</Grid.Resources>
<Slider Style="{StaticResource SliderStyle1}" Orientation="Vertical" HorizontalAlignment="Left" />
</Grid>