How do I save this plant?

My grandmother is dying and this is one if her plants that I am trying to keep alive. When i got it the white parts were wrapped in paper towels and in a cup and I tried burying them in the soil but it didn't work out well.

She stresses not to over water them like once a week and lightly. Foes anyone know what these are called? And the best way to save this one? The one pic by beavis and butthead is of a healthier one she had planted before.