this time of year, I would try three pieces of cane if you have them, pot together, put in shade, and ignore except don't let them get all dry. My theory is they can't take much wind either. Then I'd probably water and stick the crummy base in the shade too and see if it sprouts.
I read a persuasive article on Brad's Begonias some years ago in which you should put 2 nodes below surface with canes, and the underground node should make a new shoot. and this is 99 perecnt of any wisdom I think I have with begnoias.
I had one potted, and stuck a tiny spider plant in the pot, now the spider has gotten huge but it seemed to suck the begonia dry and killed it.
