When she's sick, she only takes one day off? So she shows up the next day and is sniffly? or coughing or what? Is there any chance she's calling in sick hungover or something knowing she'll get that day paid. The trip to Vegas and the age just made me wonder.
+100 OP you sound very nice but also very naive. Your nanny is taking mental health days and since you pay her for all the time she takes off she has no incentive not to do this. In fact, you have created an incentive for her to take the days off.
Someone with a "weak constitution" would be out for several days not a day every month.
You have 2 options.
1. You could just fire her. If she's doing this, she is manipulative and she will continue to screw around with you anyway. She doesn't have a good work ethic and you'll be dealing with something else. Since you are nice and naive, she's figured this out and it will be more difficult for you to spot problems before they've gone for a long time.
2. You start being an employer who is direct rather than someone she can take advantage of every month. Tell her that she has used all her sick time and any further time is unpaid. You can let her know that a continued pattern of absences even if unpaid will result in termination.