If the TV is getting the cable feed directly, then there should be an 'AUDIO OUT' from the TV. Probably a DIGITAL AUDIO OUT connection. This digital audio connection is what will carry audio from the TV tuner to the receiver.
Throw the HDMI cable away, it won't be used. Okay, don't throw it away, but seriously, it is entirely pointless and needless.
You have to hook video sources to the receiver to send it to the TV. Since the TV is the only source for cable, then you don't need to feed that video back to the receiver, you just need the audio. Feed the audio from the TV to the receiver and that's it.
Stop talking about HDMI, stop thinking about HDMI - there is no HDMI with what you have described so put the cable down and walk away.
FYI: Providing exact make/model of all products used in the setup will allow for very specific and details answers, otherwise you will get somewhat 'general' answers. But, this general answer is applicable for 95%+ of displays with integrated TV tuners.