Try adding this between your existing script tags:
<script>
$(document).ready(function() {
$('#form_message').keyup(function(e) {
var keycode = (e.keyCode ? e.keyCode : e.which);
if(keycode == '13'){
$('#send').click();
}
});
});
</script>
Note that this is jQuery code and you will need to have the jQuery library loaded -- but your existing code is also jQuery, so you probably have done that already.