To record audio, you need a 3.5mm to 3.5mm cable to capture the audio from the DS headphone jack, which sounds like is what you're currently using. But the thing is, you have to capture the audio on a different capture program like OBS Studio or Streamlabs OBS, not the DS capture program. You would plug the audio cable to the "Line in" jack on the back of the PC (usually a blue port), and then in OBS you would add an auxilary audio source to capture the audio from the "Line in" jack. Then you add the video sources (the DS capture program windows).
Essentially both OBS and the DS capture program have to be running to capture the audio and video of the DS, and OBS is used to merge them into one single stream.