There are many forms of J1, the AuPair category is a 12 month visa with the option to extend a further 12 months.
Staying on once you terminate the arrangement is not allowed, but unless the host family or sponsor reported that, it's unlikely they know. Taking on other work on an Au Pair visa is also regarded as a serious offence. If you tell them that, you will likely be banned entry to the US for a time, perhaps even permanently.
OK. With that information I'm now considering canceling the whole trip.