Stuart,
The LinkedIn button is 'on' by default when you enable the Content plugin. The other social network buttons are optional. I can see how that's confusing though.
As for your issue, if the button doesn't show, it's likely that your Javascript or Integration Domain settings in your LinkedIn application aren't correct. Check that the URL's you enter there are exactly right for your site, including (or excluding) the www. If you can access your site at both
www.site.com and just site.com (non-www), you'll need to setup a redirect from one to the other. Otherwise, LinkedIn will think both URLs are different sites, and some of the social features won't work on one of them.
Hope that makes sense. If that doesn't seem to help, please post (or Private Message) a link to your site and we can investigate further.
Thanks,
Alex