Please keep in consideration that the Intel® HD Graphics is a graphics processing unit, integrated into the processor. The ports are not part of the graphics processor, therefore if a port does not work, the issue will be related to the motherboard, Video BIOS or the driver installed (if it only fails when the operating system loads).
Try enabling Extended mode or clone display following the information on the following site:
Try also updating the BIOS of your computer, test and see if you get any video display through the HDMI display while using your system in the system BIOS. Check if the system detects the TV leaving the cable and TV connected and restarting the computer, and also try it disconnecting and reconnecting the cable while using the video driver user console.
Be sure to be using a direct connection and avoid the usage of any converters, adapters or dongles.