To verify if web testing extension installed:
1. Right click on the Test Execute icon in the taskbar.
2. Select option to 'Install Extensions'.
Check for install of any other extenions you think you might need while you are in there.
If you have a floating license free then install Test Complete on that same machine and run the test. It takes a few minutes to install but if it runs in TC then you can reasonably assume an issue with your Test Execute setup. Also, if it fails in Test Complete on that machine you can debug it.
You also mentioned Jenkins. Are you logged into the machine when the test is running? I assume you are not and the slave is running 'headless'. If you are not logged in be careful which method you use to launch GUI tests on the slave. I'd recommend running the Jenkins JNLP (web start) agent for running GUI tests.To quote the Jenkins manual 'If you need display interaction (e.g. for GUI tests) on Windows and you have a dedicated (virtual) test machine, this is a suitable option. Create a jenkins user account, enable auto-login, and put a shortcut to the JNLP file in the Startup items (after having trusted the slave agent's certificate). This allows one to run tests as a restricted user as well.'
See Java Web Start section of this page:
https://wiki.jenkins-ci.org/display/JENKINS/Distributed+builds#Distributedbuilds-Differentwaysofstartingslaveagents