You would need to plug the red and white cables from a regular Wii A/V cable into that adapter. You can set the Wii to output video w/HDMI and audio w/the A/V cables.

If you do go this route, you may find the audio to be too quiet as you are taking the sound from the back of the console without amplification. Results may vary, I had a similar setup on a PS3 and it was just loud enough for me. If it does end up being too loud for whatever reason, you may need to add a final volume adjustment tool so you actually can control it. One and two examples that I found.

It's a jank setup but it should work.