I have recently installed a Cisco ASA5508-X running V9.8(2). I added the RDP plug-in and have the same issue.
When a clientless user connects using TLS1.0 the RDP sessions work using ActiveX within IE.
However when the user connects using TLS1.1 or TLS1.2 they see the links for the RDP bookmarks. When they click on them they see the pop-up saying a website is trying to start a remote session however when this is acknowledged the session never starts, the messages "Reconnect with JAVA" and "Loading..." stay at the top of the screen but no RDP window appears.