The Badge is an element with class set to 'm-badge'. A button with a badge is defined as:
<a id="btn" href="#" class="easyui-linkbutton" style="margin-left:20px">
Button<span class="m-badge">5</span>
</a>
To update the badge you should update the button's text value.
$('#btn').linkbutton({
text: 'Button<span class=m-badge>2</span>'
});