My explanation may have been a bit confusing. I can get my PS3 to output audio to the speakers without any problems, all while having the video go through HDMI. That's fine and all the adapters must be working correctly for this to happen.
My TV has two audio outputs, an analog LR stereo and an SPDIF. There is no 3.5mm out. My speakers have a 3.5mm in, since they're PC speakers. I figured LR stereo from my TV to the speakers should work, since it works for my PS3. But no sound is coming out. All the sound settings for the TV are only related to the TV speakers. There's no way to "choose" where to send them, because if you have something plugged into the out, it's supposed to automatically go there.
I had been ok with listening to cable TV through the TV speakers, but I just got a Wii U and can't do the same trick as I did with PS3. Wii U doesn't let me split the audio out through regular A/V cables while still using HDMI for video. So that's why I'm trying to get my TV to output to the speakers, that way I could listen to both TV and Wii U over them.
I figure either the signal from the TV is somehow incompatible unlike the PS3's or the output from the TV is broken. I don't see what other possible explanation there could be. I'm debating whether I should get a male LR stereo to female 3.5mm jack, an SPDIF converter, or entirely new speakers to try to get this working, unless someone has another idea.