You can get the html as follows:
var myHtml = $('#summernote').summernote('code');
Example (not complete, will need editing)
<head runat="server">
<script language="javascript" type="text/javascript">
function funcMyHtml() {
debugger;
document.getElementById("myHtml").value = $('#summernote').summernote('code');
}
</script>
</head>
<body>
<asp:HiddenField ID="myHtml" Value="0" runat="server" />
<asp:Button ID="Button1" runat="server" OnClientClick="funcMyHtml();" Text="Get my html" onclick="Button1_Click" />
</body>
string myHtmlNew = myHtml.Value;