When I connect the headphone out socket on the computer loudspeaker to the microphone input of the tape recorder, noteworthy files make dismal recordings.
Is it possible to fit something in between to get a good recording? If so where do I get it?

If your tape deck doesn't have line-in jacks (as is the case with many low-end "blasters"), you can get an attenuator cable from Radio Shack that reduces the level by about 40 decibels, allowing you to patch from your Line Out (or speaker/headphone out) on your soundcard to the mic input of the tape recorder. Still won't give great recordings, but at least you won't have all that clipping distortion.