ng-show not updating DOM element

What you have is not idiomatic angular. It is important to stick to conventions when working with angular. Anything that is connected to the DOM should be attached to the $scope object:

angular.module('app.controllers', [])
.controller("HeaderCtrl", function($scope) {
  $scope.HelpButtonVisible = false;
  $scope.HideHelpButton = function() {
    $scope.HelpButtonVisible = false;
  }
  $scope.ShowHelpButton = function() {
    $scope.HelpButtonVisible = true;    
  }
});

View:

<button id="help-button"
        ng-show="HelpButtonVisible"
        class="button button-stable icon-left ion-information-circled" 
        ui-sref="login" 
        ng-click="HideHelpButton()">Help</button>

You need to be using ng-model, also you cant bind to primitives like that.
controller:

this.HelperObj = {"ArrofHelperObjs":[/*YOUR OBJECTS HERE*/]};
this.ToggleHelpButton = function(obj){ 
     var package = this.HelperObj.ArrofHelperObjs.length;
     var handl = this.HelperObj.ArrofHelperObjs; 
     if(package == 0){
       handl.push(obj);
       return 200;
     }
     for(var x=0;x<package;x++){
        if(handl[x] == obj){
          handl.splice(x, 1);
          return 200;    
        }
        if(x == package-1){
          handl.push(obj);
          return 200;      
        }
     }
     console.log("Error: handl= "+handl);    
     console.log("Error: obj= "+obj);
     return "Error: Conditions were supposed to have been met";
     };

then html

<div ng-repeat="x in myctrl.HelperObj.ArrofHelperObjs" >
<button ng-model="x.selected" ng-click="myctrl.ToggleHelpButton(x.html); x.selected = !x.selected" ng-show="x.selected">
 {{x.html}}
</label>
</div>

Two thing to point out. ng-model is how you bind to your data. When using ng-model, DONT BIND TO PRIMITIVES. It took me a while to get that in my head. In the example shared, my objects have the properties selected and html. ie.

 this. package = {"selected":"false","html":"tagone"};

This would work in some cases, but most of the time you are going to have more than one object. Looks like this.

    this. package = {"ArrayofObjects":[]};

inside those brackets[] goes all your objects. Like this

    this. package = {"ArrayofObjects":[{"selected":"false","html":"tagone"},{"selected":"false","html":"tagtwo"}, {"selected":"false","html":"thirdtag"}   ]};

Once you are set up like this, make a function that alters any of those values. That is what my ToggleHelpButton method is doing. I am changing the value of the selected property, of that element. Then in your html, use

<toggledelement ng-model="ctrl.package.ArrayofObjects[x].someproperty">  

x is the index of the array of objects.
someproperty is literal selected. you now have an element that is bound to a truthy or falsy value. Have Fun!

Try applying this to $scope

var app = angular.module('ido.controllers', []);

app.controller("HeaderCtrl", function($scope) {

    angular.extend($scope, {
        HelpButtonVisible: false,
        HideHelpButton: function() {
            $scope.HelpButtonVisible = false;
        },
        ShowHelpButton: function() {
            $scope.HelpButtonVisible = true;    
        }
    });
})


<button id="help-button"
        ng-show="HelpButtonVisible"
        class="button button-stable icon-left ion-information-circled" 
        ui-sref="login" 
        ng-click="HideHelpButton()">Help</button>

Always attach events and values to $scope. You only have access to the $scope.

When you use the scope, there is never any need to use .$digest() or .$apply(). Only when you break out of the $scope.