OK, there are a few simple things we can do to troubleshoot the web transactions here.
First, I would recommend enabling the trace log file for RD Web Access (RDWA) on the server. There are directions on how to do this in "%systemroot%\web\rdweb\web.config". Please set the "TraceTSWA" switch to 4 (Verbose). The contents of the trace file will give us an idea of what is happening during connection setup. If we're lucky it may point directly to a specific issue.
Second, I would recommend taking a look at the IIS access logs (which can usually be found in "%systemdrive%\inetpub\Logs\LogFiles\w3svc1"). This can help highlight some failures that can occur before the request reaches the RDWA code.