Your HP TouchSmart's integrated Bluetooth adapter creates a Personal Area Network, or PAN, for connecting business devices such as a printer, cell phone or headset ...
If Bluetooth driver is missing or gets removed after a Windows Update, add the legacy driver from the Device Manager, install the correct driver from your manufacturer, etc.
Your Bluetooth isn't broken.
Once you're in the Bluetooth settings, select Add device (or the closest equivalent). Your headphones/earbuds should appear there. Select them, and the TV should indicate that the headphones are now ...