Is there any way to add an element that is an Angular directive with jQuery methods like append() and have Angular do its compilation/linking to make it work as         
        
Ideally you should avoid doing that.
However, if you really, really, really need to, then you can inject and use the $compile service followed by an element.append.
If your directive doesn't need access to a specific scope, then you can even assign the $compile and $rootScope service to window in the run function of your application's module and then use them from outside the angular context by creating a new scope ($rootScope.new()) and wrap the appending of element by using $rootScope.apply().
The accepted answer didn't provide a complete example, here's one:
https://codepen.io/rhinojosahdz/pen/ZpGLZG
<body ng-app="app" ng-controller="Ctrl1 as ctrl1">
</body>
<script>
angular.module('app',[])
.controller('Ctrl1', ['$scope', '$compile', function($scope, $compile){
    var vm = this;
    vm.x = 123;
    $('body').append($compile("<hola x='ctrl1.x' />")($scope));
}])
.directive('hola', function(){
    return {
        template: '<div ng-click="vm.alertXFromGivenScope()">click me!</div>'
        ,scope: {
            x : '='
        }
        ,controller: function(){
            var vm = this;
            vm.alertXFromGivenScope = function(){
                alert(vm.x);                
            };
        }
        ,controllerAs: 'vm'
        ,bindToController: true
    }
})
<script>
You really want to avoid doing any jquery if you can, but I ran into a similar problem not too long ago, here was my question and the correct answer that should be able to help you out. The short answer is using $compile.
A complete example, from the Angular documentation:
// Angular boilerplate
var app = angular.module("myApp", []);
app.controller("MyCtrl", function($scope) {
  $scope.content = {
    label: "hello, world!",
  };
});
// Wrap the example in a timeout so it doesn't get executed when Angular
// is first started.
setTimeout(function() {
  // The new element to be added
  var $div = $("<div ng-controller='MyCtrl'>new: {{content.label}}</div>");
  
  // The parent of the new element
  var $target = $("[ng-app]");
  angular.element($target).injector().invoke(function($compile) {
    var $scope = angular.element($target).scope();
    $target.append($compile($div)($scope));
    // Finally, refresh the watch expressions in the new element
    $scope.$apply();
  });
}, 100);<script src="https://ajax.googleapis.com/ajax/libs/angularjs/1.2.23/angular.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div ng-app="myApp">
   <div ng-controller="MyCtrl">old: {{content.label}}</div>
</div>Try this
angular.element($("#appendToDiv")).append($compile("<my-angular-directive />")($scope));
The right way to go is to use: $compile and in case your directive returns: directive definition object (which is btw. the recommended way to go) you can then call link function on it (to inject scope, for example).
$('body').append($compile("<my-angular-directive />")(scope));
scope.$apply();