If memory serves me well, Jane had something to say about the religious reform that Henry VIII was carrying out. Henry didn't like her intervention and reminded her what had happened to his previous wife for being too independent, so to speak, cutting her short. Basically, Henry VIII would keep her quiet (or else). She only became some kind of a myth for Henry after her dead and quite some time after her death.
So, her role if she survived? Some kind of figurehead, perhaps having some influence in advancing her family and rising his son, but, overall, she had little to say.
Keeping in mind Henry'age and physical degradation, perhaps he manage to have two more sons/daughters with her, further reinforcing the succession in case that Edward VI ends as IOTL,
Not really, the thing that made Henry remind her about Anne's fate was the Pilgrimage of Grace, which is only partly because of religious reform, and her trying to convince him to restore Mary's legitimacy.
The truth is we don't know what could have happened had Jane survived to have more heirs.