I have been using bootstrap for a while, and I came across an issue when trying to add a dropdown menu dynamically.
This is the JavaScript that I have:
$
Here is a working jsFiddle: http://jsfiddle.net/szx4Y/85/.
FYI, your jsFiddle didn't work because you didn't include the Bootstrap script file. So if you were doing any testing there that wouldn't have worked.
I changed your code to call $(".dropdown-toggle").dropdown();
immediately after you append your HTML. By calling that method you are setting up the elements to handle all the Bootstrap functionality (click events, etc); there's no reason to call it every time you click your new element.
HTML
<ul class="dropdown-menu" role="menu" aria-labelledby="dropdownMenu" style="display: block; position: static; margin-bottom: 5px; *width: 180px;">
<li><a tabindex="-1" href="#">Action</a></li>
<li><a tabindex="-1" href="#">Another action</a></li>
<li><a tabindex="-1" href="#">Something else here</a></li>
<li class="divider"></li>
<li class="dropdown-submenu pull-left"> <a tabindex="-1" href="#">More options</a>
<ul class="dropdown-menu" id="mainMenu">
<li><a tabindex="-1" href="#">Second level link</a></li>
<li><a tabindex="-1" href="#">Second level link</a></li>
<li><a tabindex="-1" href="#">Second level link</a></li>
<li><a tabindex="-1" href="#">Second level link</a></li>
<li><a tabindex="-1" href="#">Second level link</a></li>
</ul>
</li>
</ul>
<button id="clickHere">Click here </button>
<div id="appendHere"></div>
Javascript
$(document).ready(function() {
$("#clickHere").click(function(){
$("#appendHere").html("<div class=\"dropdown\" style=\"display:inline;\"><a id=\"drop1\" href=\"#\" class=\"dropdown-toggle\" data-toggle=\"dropdown\"><b class=\"caret\"></b></a><ul class=\"dropdown-menu pull-left\"><li><button>Analyse Now</button></li><li class=\"divider\"></li><li><button>Analyse Later</button></li></ul></div>");
});
$('.dropdown-toggle').dropdown();
});
CSS
.dropdown-menu {
float:right;
}