In the past the eligibility for donation and for receiving a transplant were strict. These days transplant surgeons have extended the criteria for persons who can be donors, and who can be recipients. Not all hospitals follow the extended criteria, but I think your mother can be a donor for several organs.
You should already be in touch with a major transplant centre near you. Most transplant centres should have a transplant coordinator (by law, actually). Talk to the transplant coordinator nearest you, and note the telephone numbers in case of emergency. Also note the documentation that should be filled in.
The donation cannot take place at home.
Hospitals in Chennai: there should be quite a few. You could try Global Hospitals in Chennai.