Myrabeth, I find this completely bizarre. On so many levels. You handled it just brilliantly. I am trying to imagine how I would respond in this situation. Not nearly as eloquently - I would be gobsmacked.
I occurs to me that this was a financial motive for her, ie she could pay you less than she would earn on her shift???
Ha, it makes me chuckle to think of her asking you to look after her baby. Clearly she is not an intuitive person :-)
Good job.