Sometimes ya gotta stop thinking and just isolate the problem.
Before starting, before pluging and unplugging any connectors, ALWAYS make sure ALL power is off to ALL units.
First, make sure the adapter cable is working as you think it is.
Try using that adapter with a walkmans headphone output. That should do a right/left split with no problem.
Play the walkman (via your adapter) thru the aux input of a known working stereo to be sure it's working.
If it doesn't work then odds are it's the adapter that's at fault. Replace it by buying a 1/8" stereo mini to 2 RCA adapter at RatShack. That should fer sure cure the problem. But, just for S & G, try it thru the stereo again just to be sure.
Now you have the walkman playing thru your stereo, you have something to work with.
Now, let's check out that mixer by unplugging the RCA plugs from the stereo and plugging them into your new mixer. Since we know we have signal up to the mixer's inputs there's several reasons why it could work through the stereo and not the mixer. Either the mixer or something after it has problems.
Trace it out. Does the mixer have meters to show it it is getting a signal? How about a headphone jack? If the meters show a signal and it works at the mixers headphone jack but not to the amp, I'd be looking carefully at he outputs of the mixer and where the signal goes from there. If it doean't show on the meters or you can't hear it in the 'phones, it's most likely the mixer causing your headaches.
Once you get both channels from the walkman, through the adapter to the red/white RCA's, working through the mixer and the speakers, there's only one reason it won't work with the sound card. The sound card either has issues or you misunderstood the output connections. On this, my friend, you're on your own.
FWIW, red is generally the right channel, white would be the left and whatever other color remains would be the common.