In order to do this, you would need to upload an image matching the style of the ones already used for the social icons in the header of the Splendio theme. Here is an example: http://s0.wp.com/wp-content/themes/pub/splendio/images/syn-facebook-ico.png
Then you would need to add HTML to a text widget that matches the existing HTML for the current icons, except with LinkedIn details:
<li style="opacity: 1; ">
<a class="twitter-link" href="https://twitter.com/#!/bridgeadvantage" title="Twitter"><span>Twitter</span></a>
</li>
You would also want to add CSS to pull in the background image. Here's an example, you would need to use a full URL for the url() value.
#header-auxiliary .syndicate li a.twitter-link {
background: url(images/syn-twitter-ico.png) no-repeat left top;
}
Then you would need to add CSS to move the text widget contents up into the right place--which can be accomplished using absolute positioning.
This isn't really a simple task given the constraints on WordPress.com combined with how this particular theme was designed, but it's doable with some work.