I was able to do this by hacking the widgets.php file. Make a backup of the existing file and try it on a local server to see if it works.
Open widgets.php and search for the function wp_widget_recent_entries. Move down a few lines until you see a line called
<?php echo $before_title . $title . $after_title; ?>
My file had that at line 919. Then insert the following code on the next line, before the line that just has
<ul> in it. What you insert will be hardcoded to your site's RSS feed, so you will need to change the code accordingly.
<a href="http://thenameofyoursite.com/feed/" title="RSS Feed for Blog Entries" class="feedlink"><span>RSS</span></a>
Then save the file and see if it suits your needs.