First of all, if you feel that you can wait a couple more months until spring arrives, I think you should do that. There are some green leaves, even on the stem that is doing so poorly, so I think you can safely wait that length of time. The whole process will go much more smoothly if you wait until spring.
Then, take a clean, sharp knife and cut completely through the cane, so that there is just a stub about 5" long sticking up out of the soil. (Yes, cut the one that's leaning off to the side, the one with all the brown/dried leaves.) I would use a kitchen knife, probably just a paring knife, but you can use any knife you're comfortable with. Wipe it with rubbing alcohol before you cut, so the wound won't be accidentally contaminated.
Brown leaf tips can be the result of fluoride in the water. Dracaenas are especially sensitive to it. It doesn't have to be added to the water, it can be a naturally occurring mineral. The plant will still be sensitive to it, and it will cause the leaves to look brown and burned. Sometimes, just the tips and edges of the leaves turn brown, but if the amount of fluorine in the water is high, the damage can be more extensive. Fluorine is not very stable, and it will escape from the water if you leave the water sit for 24 to 48 hours in an uncovered container. (I just let water sit in wide-mouth jugs on my kitchen counter until I need it for my plants.) This technique also rids water of excess chlorine.
Another thing that can cause browning of leaves is inadequate humidity, as I mentioned in my previous post.
I would stake the leaning stalk, using a thin length of dowel, or something similar. Don't use anything too thick, as it would disturb the roots too much. Tie the stalk to the stake. Then, add about an inch of fresh potting mix to the container. That amount of new soil won't harm either stalk, but will allow the plant to form more roots, giving it a sturdier base to hold its stems erect. When the stub sends out new top growth, you can tie it to the stake as needed. In their natural setting, the stalks of these plants flop onto the ground and form roots and send up new shoots. It's one of the ways they propagate themselves. Since big plants like these look awkward and ungainly when they flop over the edges of their containers, however, they are usually staked to hold them upright.
To maintain the plant, raise the humidity, either by placing the container on a humidity tray -- usually a shallow tray filled with pebbles. The tray should be the same diameter as the widest part of the plant. Another method is to mist the leafy part of the plant with water several times a day. Allow any water intended for the plant to sit long enough to remove any fluorine and chlorine, as described above. That should resolve most of the problems I can see that the plant has.
We haven't discussed how much or how often you water the plant. We can talk about that if you like, but I am under the impression that the plant did well while in your office. That leads me to believe your watering technique is fine. Is that correct?
Check the plant for insect infestations, especially spider mites and scale insects. Spider mites like warm dry conditions. They don't like high humidity. You can usually tell they're present by the telltale strands of webbing they create. If they've infested your plant, though, you'll need to get rid of them. They suck the sap from the plant, and can cause dead and drying leaves. Scale looks like little round bumps on the stems and undersides of the leaves. They suck the sap, too, and are a bit more difficult to get rid of, but it's important to help the plant in every way you can.
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams