You will probably have a very difficult time finding your "dream nanny." That said, you can help your main nanny become a dream nanny, but it may take a lot of patience and perseverence. As several others have noted below, you need to be very clear, precise and concise about what her job entails. IMO, it is not at all unreasonable to ask your nanny to clean up the kids areas (basically, any area that the kids touch, including their laundry). And clean up doesn't mean just put away things -- it means seriously clean, wipe, scrub, disinfect, whatever. The problem is that if she just doesn't see something as being dirty, then she thinks she's done. So, that's where you will have to be really specific about what YOU think is clean and how you need her to clean such on whichever days.
As for potential illnesses, fever, bowel movements, etc., it is also not unreasonable to have your nanny watch for those things and report back to you. You need to know whether you child is getting sick (seems obvious) or is having trouble with BMs or diarrhea, etc. It is definitely part of the nanny's caregiving job to help the children stay healthy. Also, IMO, I would not be happy if my son's nanny was sleeping on the job unless I told her that that was an acceptable part of her job. When your kids are napping, she should be doing all the other things that you have requested (cleaning, laundry, etc., for the kids).
Now, for the positive, the fact that your nanny plays (and enjoys playing) with the kids is really important. You want your children to be stimulated (and not by the TV) and encouraged to play, read, be active. That's a huge plus. I'm sure you're paying a premium for your nanny, but she may also be worth it. On the other hand, if you enroll the kids in preschool (which I highly recommend), then preschool will provide a lot of acitivity and education playtime, and so you may not need such playtime with your main nanny.
In any case, I know I didn't address all of your questions, but I just think you need to be very explicit about what you want/need and give your main nanny a chance to correct any errors, misperceptions, etc. My personal opinion is that she ultimately will not be able to meet your standards and requirements, at which point either you will have to bend your requirements (this may be like trying to change your nature) or find a new nanny. I'm guessing your weekend nanny may not be able to fit the bill (legal to drive and pick up the kids from school?) in the long run, but it sounds like she is worth trying out for weekly duty, particularly if the kids are in preschool.
In my personal experience, we had a cleaning person come to our home once every 2 weeks, but our cleaning person was so unreliable that we let her go. Since our child had just started preschool, our nanny volunteered to clean the entire house during preschool time (9-noon), and then take care of our child after school. Even though her hours didn't change, we still gave her a 10% raise to cover the additional work. We really love our nanny and everything has worked out very nicely (even though we still have the occasional bumps in the road...). Best wishes.