After playing around with this feature, I noticed that the HTML looks like this:
<iframe id = "Some iFrameId">
#document
<html>
<head>...</head>
<body>The text of the editor</body>
</html>
</iframe>
In the ASPX, I did the following to make my life a litle bit easier:
<div id ="myDiv" ClientIDMode="Static">
<ajaxToolkit:Editor ID="Editor" runat = "server" />
</div>
By doing this, I have simplified my problem to become find the iFrame enclosed in myDiv, which contains the HTML of the editor.
To do that in the JS
//get the iFrame
var myIframe = $("#myDiv, iframe") //this returns an array and myIframe[1] is the iFrame, which contains the text.
//get the HTML from the iFrame
var content = myIFrame[1].contentWindow.document.body.innerHTML;
Now content contains, what I was looking for. This is kind of long, and there might be an easier way, but after searching for a solution, I found most of them to be:
do a .get_content or some function call, which didn't work for my case.