I used some rather small pieces of duck tape to hold the shoelaces in place. It worked well. Notice I used the day-glo red tape to help with the activity illustration. You can use masking, electrical, probably almost any kind of tape.
I found it best to:
Wrap the piece of tape around the end of the wire.
Lay the lace on the tape.
Wrap the lace and tape around the wire until the tape was fully wrapped.