If you want to set this for just one tooltip, set the duration on the object having the Tooltip, like this:
<Label ToolTipService.ShowDuration="12000" Name="lblShowTooltip" Content="Shows tooltip">
<Label.ToolTip>
<ToolTip>
<TextBlock>Hello world!</TextBlock>
</ToolTip>
</Label.ToolTip>
</Label>
I'd say that this design was chosen because it allows same tooltip with different timeouts on different controls.
If you want this globally for your whole app, see the accepted answer.
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