Sometimes God has to lead us through some hard times before we'll listen to His voice; perhaps this was true with you. Ask God to use you to help others find Christ who also are going through hard times, just as He used this woman in your life.
Often the hardest people to talk to about Jesus and what He has done for us are the members of our own families. One reason, I think, is because they know us so well, and we know they will be watching for any inconsistency or hypocrisy. We know they also will be put off by any hint of superiority or pride (as you have suggested). But this doesn't mean we should keep silent! After Andrew met Jesus, the Bible says that "The first thing (he) did was to find his brother Simon and tell him, 'We have found the Messiah'" (John 1:41).
Pray for your family every day, and ask God to open up opportunities to share with them one-on-one how you came to know Christ. Admit that you know you aren't perfect, but tell them honestly what a difference He has made in your life, and how you now have the hope of heaven because of Him.
In addition, as Christmas approaches think of ways you can tell them what it now means to you. For example, send Christ-centered Christmas cards this year, and include a brief letter about your cancer and how God used it to bring you to Christ.