Dinny, I'm not one who usually chimes in on these matters, but...
First off, I'd suggest leaving the short baseball bat on the playground. A long bladed kitchen knife ( butcher knife, thick blade ) is much more intimidating, and deadly. Hopefully find one with a hole at rear of handle to make a short loop that she can run her wrist through ( para cord works fine ). Spend some time with her using a paint stirring stick ( ruler?), teaching her short underhand jabs, standing quartering away from opponent, with weak hand forward ( grasp..push..open/close doors, etc ), strong hand ( knife ) facing forward, alongside body.
As for first weapon, I simply cannot think of anything better then something like the Ruger LCR. Something she can keep with her when she answers door, walks outside, etc. etc. A good flashlight ( like a small surefire, etc., not one of the old long kel-lite jobs.
Mostly it comes down to mind-set. She must be of the mindset that she and/or family WILL NOT be a victim.
Mark