$20-$25/hour is pretty standard for babysitting in NYC; often an uber ride home - there's no shame in using a shared uber pool to get the siiter home at a fraction of the price;
I don't think the Barnard Babysitting Service applies for you - they're looking for semester long recurring gigs;
you're on the Upper West Side - Barney Greengrass is probably worth going to for breakfast one day - try to avoid the lines on the weekend for brunch and order the potato pancakes as an appetizer for the kids when you sit;
on the Upper East Side we like to take the kids to Via Quadrono for lunch or an early dinner after a museum trip; Sant Ambroseus works as a more expensive version of this as well;
Amazon Prime Now is great for delivery of whatever staples you'd like to your hotel or residence.