K.J.
A., I believe that you should be honest with her about why you must find a new nanny. It may be difficult for you to do, as well as for her to hear, but you will be doing her a great service. As a teacher, I find that in life you must always be looking to help instruct people who seem to need it. It extends outside of the classroom and this girl clearly needs to be directly told that her lazy manner and inability or unwillingness to change are what has led you to replace her. Hopefully she can learn from it and when future families tell her that they are looking for more out of her, she will get with it.
I don't think it takes any special training to come up with structured activities and projects. If someone is advertising themself as a nanny, such things come with the territory. If she WANTED to, she could easily Google some activities to do, or pick up a few little books on kids' projects/activities--like the rest of us do.
I also do not think that you should give her a 2 weeks notice. The 2 weeks severance is more than sufficient. After all, it is the income during those 2 wks that is the important part for her, and it sounds like you should really move on asap.
Good luck!