If you are really interested in making a division from the cane your cat cut down (if there are NO roots on the plant), then you should cut right below the node, and it may kick up a new growth in Sphagnum moss or even in dirt (need air circulation though), then it might not. If it was to sprout a new growth, it would take a VERY long time for it to get to a substancial size or even bloom.
If the cane has roots at the bottom, then just pot it in a finer medium, and keep it watered almost every day...
Is the broken cane a newer growth or an old back bulb? If it is small compared to the first blooming sized growth, then it's not worth the time and energy to save it... hope some of this helps,