Problem scenario
With an HDMI cable you cannot get an external monitor that is connected to your Windows computer to work. In the Control Panel section for "Screen Resolution" there is a drop down menu for "Display." This will give some descriptive identification of your monitor when the monitor is working properly. It does not show an attached monitor. What should you do if the "Display" drop down in Screen Resolution (in the Windows Control Panel) does not show your monitor and the external monitor is not working to display any output?
Possible solution #1
Can you test the monitor and cable with another computer? The cable could be bad.
Possible solution #2
Assuming the monitor and cable work, open File Explorer and browse to "This PC". Right click "This PC" and go to "Manage". Go to Computer Management -> System Tools -> Device Manager -> Display Adapters. Double click the item underneath "Display Adapters" (e.g., "Intel HD Graphics). Go to the "Drivers" tab. Click "Rollback Driver". If "Rollback Driver" is grayed out, try "Update Driver" -- even if you have no access to a network or the internet. Changing the driver here can enable the external monitor to work.
Possible solution #3
If you need a new monitor, you can purchase one here.