The quickest solution seems to be:
- Open Google, and search for "java plugin verify"
- Click the result with a title similar to "How do I test whether Java is working on my computer?" See the screenshot below for reference.
- Click the Free Java Download button
- Click the Agree and Start Free Download button. This should prompt you to download an installer such as "jxpinstall.exe".
- Once the download is complete, close Firefox
- Run the installer, and follow the on-screen instructions to complete the installation
- After the installation, return to the verification page to make sure the Java plugin for Firefox now works
For some reason, downloading and installing the JDK or the JRE from Oracle's website does not install the plugin correctly for Firefox. This is the only reliable way I've encountered so far of getting the plugin to work in Windows Server 2008 R2.