From the postiing you wrote, you and she seem never to have discussed who would do what and by what dates. Good lesson for next time: Meet at the very start, assign who does what at the very start, and tell the troop leader who is doing what. For now, you need to phone her today and say you have done the following things XYZ, and the remaining ones on the list -- which she has, remind her of that -- are hers to do by a certain date. But don't let her have until the date of the event to do her part, or you will find that event day arrives and you have disappointed girls and a ticked-off leader because she may well turn up having done nothing at all. If she is balky and makes excuses, just be up front: "I'm sorry we didn't get together at the start, but the event date is X and we have joint responsibility for giving our girls an event that will not disappoint them. We've known about this since (date). I have already done these things -- will you do these others by day X (NOT the event date!!) or not? Because if you can't, I cannot either, and we will have to let the leader know the situation."
One thing you may run into: You say you "began doing things on the list, hoping that the other mom would pick a few things herself." You say "that hasn't happened" but are you certain? I'd be worried that maybe she was operating on her own-- as frankly you are already doing yourself -- and SHE may have picked items to do that you have already done! Duplicating effort is a total waste of time for both of you. You and she need to confer immediately to ensure she isn't doing what you've already done, and/or to ensure that she does something toward her obligation.
She may be a last-second kind of person. That doesn't work for you or for me either, and with folks like that you really, really have to communicate both verbally and in writing -- e-mails are a great trail for you to say later, "I told you in that e-mail on this date that I was doing X and Y, and I have your reply here from (date) that you would be doing A and B. Is that done yet?"
You and she haven't communicated and -- to be frank -- that is really on your head as much as on hers. You "don't like to boss people" but it is not bossing people to keep them to their obligations. Remind her that this for the GIRLS, and they will be let down if there isn't an event or if it's badly planned.