There's nothing to be done with the piece that broke off as its no longer attached to the roots. Its wilting. But another option is to grow new trees from cuttings.
Figs root easily from cuttings just about any time of the year. The longer the cutting the better (In Europe, the cuttings are 3 or 4 ft long). Bury the entire stem with just the tip out of the ground (Cut all the leaves off). If you take the cuttings and let them callous for a week or so and treat them with 1000 or 2000 ppm IBA your chances will improve.
I've never bothered with the rooting hormone so my rooting success in warm months is less than 50% (in winter, almost 100%). I usually make my cuttings about a foot long if I'm starting them in 5 gallon pots and longer if directly in the ground. Put several in each pot.
