Which of the following diseases can be transmitted from an infected mother to her baby through breastfeeding?

AIDS, also known as Acquired Immune Deficiency Syndrome, is a sexually transmitted disease caused by the human immunodeficiency virus (HIV). Infected body fluids like blood, semen, and vaginal discharge.

Even breast milk can transmit the disease from an infected mother to her baby during breastfeeding. 

Therefore, AIDS is the correct answer. Malaria, Kala-azar, and Elephantiasis are not transmitted through breastfeeding.