Is there anyway I can come out of an HDMI port on my tv and convert it to an auxiliary port that I can plug into the base of the Pyle external speaker using the 3.5 mm jack? I bought a cable- Deconn HDMI to HDMI/DVI/VGA Audio 4 in 1 Adapter, but this seems like it wants to convert the auxiliary to the HDMI and not have the HDMI as the input and the VGA as the output to my base of the Pyle.
The HDMI ports on the TV would be inputs, you can't connect an external speaker to that. What you may be able to do is connect the speaker to whatever source is connected to the TV, cable box, disk player, etc...
The only possible way would be to connect a speaker to line level converter to the wires that connect the internal TV speakers. That will give you a variable analog audio output that you can connect to a bluetooth transmitter or directly to the speaker if it has an aux input.
If you don't use the TV tuner then you can use an HDMI input selector with audio extractor to provide an audio output from all your external sources.