First of all balanced does not always sound better then single ended and many times it does not sound as good, depends on the equipment and the matching of that equipment. If your CD player is single ended you will be wasting your time and money using a balanced cable with an adapter and unless a cd player is truly balanced which yours is not, trying to convert it to balanced will be a waste of time and surely degrade the sound since a player to be truly balanced must be balanced throughout, not just at the output stage.Better to try a higher quality single ended interconnect or just leave everything alone.
Mike
Mike