it really depends on what you want the plant to look like. If you want a short plant prune back the leaves that are growing too tall and the plant eventually only grow that tall. if you want a giant plant let it grow wild. eventually it will shoot out tiny little dwarf Lilys. I don't know of any way to encourage them to make babies. they really like root tabs and a regular fertilizer dosing like most aquatic plants. Hope that helps.
As long as the plantlet has established roots it should do just fine being cut from the bulb. The bulbs of lily are like potatoes, they will grow as many plants as they physically can before running out of food, in a rich environment the plantlets naturally let go and form massive root systems of their own.
The picture you have shows it has some good roots, you can attempt trimming the thick connecting root from the bulb and replant the roots in another location. Sometimes there will be melt going from tank to tank if the tank parameters are different from each other.