Scholars in the Jennings special said Jesus might have been illegitimate. Can they offer any proof?
No, of course there is no proof. There has long been suspicion in some quarters that a story of a conception by Mary without a husband was a simple cover-up for a dalliance on her part. There even exists a later story complete with a named Roman soldier father. In certain Jewish sources, Jesus is actually called a mamzer, a child of fornication, but these appear to be polemic texts that emerged after the divorce between Judaism and Christianity was complete.
How do you reconcile your Christian faith with the facts you have about Jesus?
The question assumes that these "facts" somehow challenge my Christian faith. I do not know what these facts might be. I have found long ago in discussions with my colleagues that often what are taken to be facts are simple assertions--ones that amount to speculation, assumption, or theorizing. Everyone has the right to do these three things, but that does not establish a "fact" in our ordinary understanding of the word.