Hi ,
I would suggest you do nothing except pray for your neighbor.
Obviously she does not want to share information with others. Cancer is very invasive. She must be in shock herself and probably does not want pity from others. Just pray for her. When she is ready to share with you, I am sure she will.
That other neighbor should not have share. When people confide their personal information and specifically request not to share, it is a betrayal of confidence to share, especially when one is seriously sick and vulnerable.
Be blessed.
Jockey