I don't think there is a single word to describe this. I like Juhasz' suggestion of "not funny", but it may be unclear exactly what you mean.
Instead we would say she is "bad at telling jokes" or even "terrible at telling jokes".
I have a friend who is just terrible at telling jokes. She knows a lot of jokes, but always forgets or messes up the punchline. In fact she's so bad at it, it's actually pretty funny.
The situation you describe is somewhat more generic, and might be better described using by one of the synonyms of "shy" like reserved, reticent, withdrawn, close-mouthed, or hesitant.
I'm sure she knows a lot of jokes, but she's too reserved to ever tell them in public.
If, as you say, she doesn't tell jokes because of her personality, try serious, somber, humorless, sincere, or no-nonsense.
I tell her I think she shouldn't be so serious all the time and maybe try telling a joke once in a while.
Or perhaps some variation on genuine, thoughtful, authentic, natural, earnest or even naive or ingenuous.
She's so genuine and so nice that she can't even see the humor in jokes that make fun of other people.
Still, it's not clear why someone who is "pure" would be unable to tell a joke. She might not want to tell jokes that hurt other people, but there's nothing hurtful about something like these silly kids' jokes:
Q: What did the paper say to the pencil?
A: Write on!
Q: Why was the broom late?
A: It over swept!
Q: What did one toilet say to the other?
A: You look a bit flushed.
