Yeah Soybean is right. Use your line out on the Realtek.
However, as far as using the mic input on your karaoke machine, it is possibly (probably) only a mono channel and depending on how it is wired it may only record either the left or right side of your stereo minijack cable. It would spread that one side across the whole stereo spectrum, but you will still only be hearing a mono recording of one side of the original stereo mix from the computer. I guess you could get a cable that is 1/8 inch minijack stereo on one end (plugged into the stereo line out of your realtek) and splits into two separate mono 1/8 inch minijacks. You could plug each of these into the two mic inputs on the karaoke machine as long as you keep them recording into the karaoke at the same volume. Meaning if each mic input has a gain/volume knob, keep them both identical. This way you would have a mono recording still, but it would have both sides of the stereo mix blended together. May not sound the greatest though.
There is maybe another and better way to do this, which would be to get the 1/8 inch minijack stereo cable on one end (plugged into the realtek) that is RCA on the other end. Then plug the RCA end into the back of the karaoke unit's AUX inputs (page 4, input "40", called the AUX IN Left and Right Jacks,
http://www.memorex.com/manuals/MKS8503.pdf). You will have to test to see if it records the AUX in to the tape though, as the manual does not say. If it does, then you'll be able to retain a stereo recording to tape. If not, you may be limited to a Stereo summed to mono recording.
If the stereo CD recordings you are putting to tape are of high quality, you can still get a decent blended mono mix, but the method using the AUX inputs would be the best way, as long as the unit records the Aux Ins.
One last note if you plan to do this alot. If you can afford it, I would invest in a consumer or professional level cassette or dual cassette deck. I won't recommend any as there are too many to choose from, but you can easily research on the net depending on what you are looking for. I guess the only important thing in this case is to make sure you have a cassette recorder that can record with a stereo input if you want to maintain the music recording on tape in stereo.
If you need clarification on anything let me know.