Don't be afraid to take cuttings of things. Most plants respond well to occasional trimming and it helps them bush out if they are looking scraggly. I haven't attempted cuttings of my Mimosa, but from what I hear they aren't too difficult.
I'd use a very light rooting medium that holds some water, like perlite or pumice and put them under a humdity dome. Take your cutting and remove almost all the leaves. Stick it in the moist substrate, cover with whatever you are using for a dome and put them in a warm spot with moderate lighting. Rooting hormones might help, but having never tried Mimosa I can't say how it might respond. Some plants don't really respond to hormones, and with others it's easy to overdo it.
Some plants root better from woody cuttings, some from green wood cuttings. I don't know which is better for Mimosa, but a little research would no doubt get you pointed in the right direction.
Experiment and find out what works for you. AND let us know what that turns out to be!