jquery replace html and javascript function

UPDATE

According to the changes in the question the new problem is the position where you declare the new value for the variable a.

Because:

jQuery(document).ready(function () {

executes the code when document is ready, and because your document is already ready when you substitute the html of hello element, on the first click you get the value 5 and only after you get the new value. From the second time, because the html is changed, when you click you activate just the ready function. The solution is to declare the variable a in the data string immediately before the document ready.

You may use the added button "describe function" to see by yourself the content of the script contained in the hello div.

To better understand I added a string as a parameter to your func function in order to understand how and when it's called.

Updated snippet:

$(document).on('click', '#btn2', function (e) {
  $('body').append('<br>var a is: ' + a + '  func() is: ' + func.toString())
  .append('<br><div style="border: double;">hello HTML: ' + $('#hello')[0].outerHTML.replace(/</g, '&lt;')
          .replace(/>/g, '&gt;') + '</div>');
});

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>

<div id="hello">
    Some text
    <script>
        jQuery(document).ready(function () {
            func("inside ready");
        });
        var a = 5;
        function func(str) {
            alert("FIRST: " + str + "  Value is FIVE. Integer value: " +a);
        }
    </script>
</div>

<div id="mybtn" style="border: 2px solid black; padding:5px; background-color: yellow; width: 100px">CLICK</div>
<script>
    var data = " \
		  Some text2 \
		  <script> \
            var a = 6; /* HERE!!!!!!!!!!!!!!*/\
			jQuery(document).ready(function () { \
				func('inside second ready'); \
			}); \
			function func(str) { \
			  alert('SECOND: ' + str + ' Value is SIX. Integer value: ' +a); \
			} \
			<\/script> ";

    $('#mybtn').on('click', function(e) {
        $('#hello').html(data);
    });
</script>
<button id="btn2">Describe function</button>

I created and tested a possible similiar situation, and it works for me:

$(function () {
  var newcode = '<script>' +
      'var var1 = 3;' +
      'function func() {' +
      '    console.log("VALA is 4");' +
      '    console.log("VALB is " + var1);' +
      '}';
  $('#btn1').on('click', function (e) {
    $.getJSON('https://api.github.com/users', function(data) {
       $('#box1').html(newcode);
    });
  });
  $('#btn2').on('click', function (e) {
    func();
    $('body').append('<br>var var1 is: ' + var1 + '  func() is: ' + func.toString());
  });
});

<script src="https://code.jquery.com/jquery-1.12.1.min.js"></script>

<div id="box1">
    <script>
        var var1 = 2;
        function func() {
            console.log("VALA is 1");
            console.log("VALB is " + var1);
        }
    </script>
</div>
<button id="btn1">Simulate Ajax Call</button>
<button id="btn2">Call function</button>

This has to do with the JavaScript parsing and execution of the new script on the page. When the function executions, when the NEW document ready triggers the func execution the OLD values are still in place.

To detail, these occur in sequence (roughly)

1. New code gets put in place (parsed by the Javascript) replacing the OLD function with the new one (with the SIX in it)
2. The NEW document ready triggers (calling the NEW function with the OLD a value still in place (the 5)
3. THEN, the a value is replaced with the NEW value of 6 (no further calls with the NEW value are executed

See here how I changed the snippet. I added two alert('a:'+a);\ in your code. NOW execute it an you see the sequence of alerts:

The SECOND one alerts the 6 after it has changed.

   var data = " \
              Some text2 \
              <script> \
                jQuery(document).ready(function () { \
                    func(); \
                }); \
        alert('a:'+a);\
             var    a = 6; \
            alert('a:'+a);\
            function func() { \
                  alert('Value is SIX. Integer value: ' +a); \
                } \
                <\/script> ";

    $('#mybtn').on('click', function(e) {
      $('#hello').html(data);
      console.dir($('#hello').html());
    });

NOTE: Move this var a = 6; \ BEFORE the document ready in the string and it gets executed first.

Think of that as if you had this:

    var  a = 6; \
    func(); \

EDIT2: document ready

Review the jQuery source; when $(document).ready() fires the first time, it will unbind from the functions that were previously registered (e.g. forgetting them). As such, calling jQuery.ready() manually (like you have) a second time does not re-trigger the same function calls again. SO you do not really need that wrapper in script loaded after it has fired on a page as it simply does an immediate execution of the code inside the NEW document ready since your NEW bound function executes then is unbound.