Do you have a two year foreign residency requirement? If not, consistent with the information given in dozens of threads in this forum, you would marry then petition to adjust your status on the basis of marriage. If so, you need to get the foreign residency requirement waived, or you need to spend those two years outside the U.S. before you will be eligible for lawful status (even through marriage) following the expiration of your J1 visa.
If you wish to work for compensation after petitioning to adjust your status, you can apply for employment authorization. But your sponsoring organization may not be willing to continue to extend the Au Pair position after you seek to adjust your status. I don't know their policies, so check with them. In the alternative, you may wish to consider waiting to adjust your status until your current Au Pair assignment is over, even if you marry sooner.

