Solution 1:

The NirSoft Network Password Recovery tool worked for me, seems like Windows now saves Remote Desktop passwords with the other network passwords. From the tool's description:

When you connect to a network share on your LAN or to your .NET Passport account, Windows allows you to save your password in order to use it in each time that you connect the remote server. This utility recovers all network passwords stored on your system for the current logged-on user. It can also recover the passwords stored in Credentials file of external drive, as long as you know the last log-on password.

Make sure you grab the x64 version if you are running on 64-bit.

Solution 2:

You may not be able to recover the password depending on how it was stored. However I may have a workaround based on a comment you made

If I could simply change the ip address and use the same (hidden) login information for this one RDP file I would be saved! :S

If the connection uses a DNS connection instead of a IP address you can put a entry in the HOSTS file to have it point at the different server.

If you are using IP you can use the windows tool netsh however you may need to enable the Routing and Remote Access service for the routing namespace to show up in netsh