Will she be using the car to take your little one for any activities, or would she be happy to do so if needed in the future? If so, I would split the cost of the permit. Our nanny doesn't drive, so for certain activities they need to take the bus, the cost of which we obviously cover. I would also add that her having her own car may be of great benefit to you in the future.
Edit: most importantly I note you said she's a great fit for your family - if paying for the permit in full is a deal breaker, then for me personally I would pay it if you can afford to, as finding someone who you like and trust enough to look after the most precious thing you have is far far more difficult!