In the email file for the checkout success page you have the order number on line 40 set like this:
<p style="color:#50516B; /* Accent_1 */ margin-bottom:7px; font-size:medium;">Invoice number: ?</p>
You should make sure to set it to the session variable like you have on the confirm page:
<p style="color:#50516B; /* Accent_1 */ margin-bottom:7px; font-size:medium;">Invoice number:<?php echo ((isset($_SESSION["eCart1_OrderID"]))?$_SESSION["eCart1_OrderID"]:"")?></p>
If you are not getting the order id to show after this then there must be resetting of the session occurring somewhere. Since you would normally go from the confirm to the checkout success page the problem would be on one of these two pages. Please give this a try and let us know the result that you get.