In my experience of nannying and charging for it, a family with more than one child would be paying more than a family with just one, and depending on the ages of the children, it might be substantially more. A set of twins aged 3 plus a baby would mean top dollar for the nanny.
I did a nannyshare job once and was paid by both sets of parents, for two toddlers at a time in the house of one family.
At another job, when a baby was born the family raised my hourly wage to 1.5 times what I had previously been earning.
Wrt the issue of resenting paying her for time you book but end up not using -- yabu there too and moreso than I originally remarked. If you have her booked until 6 pm she may have turned down an offer of a job at 6 pm or even one at 6.15 unless the potential client lived next door to you. In effect you have monopolised her time for more than just the bracket you booked her for. So yes, you need to pay her for time booked.
To those saying the sitter's work was not doubled -- in my experience as both a parent and a nanny, 5, 6 and even 7 year olds are incredibly hard work when they have a friend over. A child who is already having a hard time due to asthma plus a possibly giddy friend whose parents are going through a divorce (and who is perhaps worrying what the heck happened to her dad who was supposed to pick her up) would be a lot of work. But I agree with DrSeuss' argument - at what point does it become a case of 'extra work' when extra children are added?
BackforGood:
...just say you were a bit disappointed that she demanded extra money for no extra work, when remember you never try to pay less when she does a shorter time, and - although it works for you, it also works for her - a 'mutually good arrangement' so she might want to think about that going forwards.
In other words, it's fine for people to take advantage of teenage girls who are both available when you need them and responsible enough to take care of a child.
It's not 'no extra work' -- it' the responsibility of a child she did not know on top of responsibility for one with bad asthma at the moment.
Never trying to pay less? Given that this teenager may turn down a subsequent job that she might otherwise take if the OP got her act together and predicted more accurately when she would be home, the OP would have some cheek even thinking about paying her less. It is the commitment of time that the OP is paying for.
It certainly does work for the OP -- try finding a qualified nanny for those hours. Try paying a CM for those hours.
It also works for her? I don't know how things are where you are, but my DD3, aged 17 and with a driver's licence, has people lining up for the sort of after school work the OP is taking for granted. 3 o'clock to 6 o'clock are incredibly hard hours to fill.
Just because the work involves children and the worker is a teenage girl doesn't mean you treat her like your pet monkey.