Hi Ernie –
Unfortunately, African Violets can not be propagated by their roots. However, if there are any salvageable leaves remaining on your plant you can give it a fresh start by taking a few “cuttings”. African Violets are very good at this method of propagation but it will require about a month of patience before you see any new plants beginning to grow.
Begin by gently removing some leaves from the plant (probably as many as you can in this situation). Next, make a 45 degree angled cut at the base of each leaf stem with a sterile blade. Place each cutting in a clean beer bottle filled with bottled water. Your reservoir does not have to be a beer bottle specifically, just something that will support the leaf above while the cut stem remains under water. Keep the the bottled cuttings out of direct sunlight as they begin forming roots and check the water often for clarity and cleanliness. When necessary, change the water out. Once you notice root development at the base of the leaf stems, move your newly rooted cuttings into small pots filled with a potting soil with good drainage. Water in your new plants well and place in a sunny window. Additionally, it is a good idea to loosely place a plastic bag with a few holes over the pots to help increase humidity as they really begin to grow. After about a month you can remove the bags and you should have happy healthy new plants!
Good luck, I hope this helps!