I don't know if this is relevant or not - and I may have missed something somewhere. But, is WeirdMom not a native English speaker? Her emails are worded very oddly. I'm trying to find out if a cultural difference could be at play here.
Although, regardless, I think she's just too much and doesn't belong in your group. I find it incredibly odd that she wants SO MUCH to be in YOUR group. Honestly, if someone told me I didn't fit their group, the last thing I'd want to do is join it anyway. I'd be thinking the whole time that no one wanted me there. Are there not other groups she can join - even within the local organization itself - maybe another playgroup. Not that I think you should have sent her on to be someone else's problem, but her desperation just makes no sense to me. Just find a different group already.
You definitely shouldn't have sent the last email (the, no, really I meant it leave me alone email). But, it's not that big of a deal. Just ignore her completely from now on.
As far as spreading the news...I would approach one person in the leadership, tell them what happened, and ask if this is the kind of thing leadership would want to know about. The person will either say yes, or tell you to just let them know if she starts stalking you.
Oh, and personally - I'd tell the other members of the group something. This woman is really aggressive and it will definitely be easier to stop her from showing up (in terms of telling other members not to give out info about meetups) than deal with her if she does show up. If she shows up somewhere, I can nearly promise it will end in much yelling, arguing and name calling. This does not sound like a person who will leave quietly when asked to.