I dont think that it can be saved, sound to me like it has rotted from the damp and cold weather.

Unfortunately that spongy thing in the branches sounds like fungi, if it is grayish it is called "Botrytis cinerea" there is nothing you can do but you can still save the tree by taking drastic actions. 1] Take the leaves out by hand or cut them off, once you are left with the tree trunk check it out and see if there is any more of that spongy thing. If so cut it back until you are left with the healthy part of the tree. 2] For safety reasons take it out of the pot in order to change the soil, withdraw it from the soil and take out all the soil in the pot and throw it away in a plastic bag. 3] Replace the soil with a good quality one plus humus and plant the tree again. If the roots are very long you can cut them back 1/3 of the length. 4] Once replanted make sure it is not shaky but firm in the soil. Water it very slightly. 5] About a week after apply growth hormones made of marine algae mixed in water to the root area. Water it slightly. If it is still cold in your area pot the tree, take it inside and mulch it until the weather is nice. 6] Use the marine algae growth hormones each 15 days and according to the instructions on the box. This only till you see that new leaves are growing. 7] Water it slightly once a week and don't have the tree near any heating device. 8] Once new leaves are growing, use the marine algae only until mid Spring, then replace it with some good fertilizer which will provide the nutrients it is not getting from the sun. Spray on the plant. Try "Grow More Fertilizer" I have used it and it is great. Check this link: http://www.gardeningknowhow.com/indoor/how-to-care-for-a-rubber-tree-plant.htm Good luck!!