How do I soundproof a room behind an open doorway with little room for a proper door?
You can add a pocket door-use an outside type door that way is is more dense. Mount it on a track inside the laundry room and slide to open or close. Whatever gap is there between the wall and door cut some trim or dense foam to fill it in..if you use the foam attach it to the door...If you do not have enough wall space maybe you could use two solid pine folding doors but take off the hinges and 
 add top slides. And match or paint to the kitchen area......They will look very nice..