You didn't mention that you want a button. The search form would still have worked - by pressing [Enter] on the keyboard.
To replace the magnifier with a HTML button, replace, in searchform.php:
<input name="submit" value="Search" type="image" src="<?php echo get_bloginfo('template_directory'); ?>/images/magnifier2-gray.gif" style="display: block; border:none; padding: 0 0 0 5px; margin: 0;" />
<input type="submit" class="button" value="Search" />
The value="Search" can be any text. The whole form should adjust to the width of the sidebar. The longer you make the text the shorter the input field should get. value=" " (button without text) or value="»" (button with right angle quote as text) should work as well if there's not much space in the sidebar.