I went to school at one of these 'agencies' to train to be a nanny, but just about anyone could go thru them for a job and get lots of jobs. The thing is, they do background checks and other checks (at least they do in Oregon) to make sure that they are qualified. And they usually call the parents to ask how the person did. If they didn't do well, then they were definitely reprimanded or fired.
Since you used an agency, DEFINITELY call and complain to the agency. Tell them exactly what she did and didn't do. I don't blame you for expecting her to do all the things you asked her to do. When they work for agencies, they know up front that they are supposed to do exactly as the parents want them to do and that they are not only representing themselves but also the agency. Period. And if they don't they can, and probably will, get fired.
I was a nanny for two years and know that if I did as this sitter did, I would have been fired on the spot. Even a regular sitter has to do something other than just sit there to be sure that there are no fires and they aren't kidnapped. I have done sitting jobs and still did what the parents wanted, when they wanted, etc. I never got fired for that! Either this girl is too young not to know better, just wanted to make some easy money, or was lazy, or all of the above. But DON'T let any of those be used as excuses. She MUST know that what she did is NOT ok, ever. She put your kids in danger, kept them up way too late, let them watch things that kids that age should NOT have been watching, and was asleep on the job so anything could have happened and she wouldn't have known it.
Do talk with your children about their experience, what they saw on TV, etc. Answer any questions they may have about all of it. Then just be loving and caring and do as you always would and they should be fine. Just let them know that you are always there to listen if they have questions about any of it.