Not sure where you are at, but here in Illinois (Chicago Suburbs) the going rate for a nanny/baby sitter is $10 per hour per kid. So if they have two kids, it would be $20 per hour. These rates are for everyday nannies/baby sitters, not ones who work with special needs kids. Even though you are not necessarily trained to work with special needs kids, you do know how to help them adequately I'd guess. You are severely underpaid in my opinion and deserve compensation. Take the other job, even if it means going without work for a month. The pay will be better and you will be much happier.
Strength is not defined by physical ability. It is determined by your actions and the compassion of your soul.