The expression goes the best way to spill a secret is to tell another parson - Ned knew that secret means life or death for Jon because if Robert would have found out - it would have meant instant death or war with his best friend. This is one secret he couldnt chance on anyone even wifey.
I agree with you - she is a young girl who has to walk very softly and not piss anyone off if she wants to survive - she is of a different temperment than Arya - Sansa from a child as the oldest girl was being groomed to marry the Prince - Robert's Son

lol

and Ned's Daughter was destined from birth. Her mother taught her in the ways of a high born lady - Arya the little tomboy was allowed to do whatever since she was focused on Sansa -
The Game of Thrones is fought in the battlefield and in the Halls of power - so she will be ok especially if she marries Tyrion and he schools her.
HAve not read the books so all this is based on the show (btw nothing on the show hints at Jon being the son of Ned sister ) - the book readers are leaking over to the show thread and getting spoiler-y