Me and my girlfriend came in to US in 2009 with J-1 visas and stayed undocumented for five years.

Based on those facts alone, I am truly sorry to opine that it is very unlikely that you will be granted a nonimigrant visa. Although you might not have acquired unlawful presence due to your J-1 status, your status violation is of such length and nature that I would venture would make it impossible to overcome 214(b) immigrant intent presumption. There are a fw nonimmigrant visa classification for whic you might qualify, but you sill need a lawyer to explore t hem with you.

It sounds like you both triggered 10 year bars from the US. To return prior to that you would need a waiver based on extreme hardships to a US citizen relative and you would need a qualifying relative to petition for you (your children cannot do so until they are 21). There are some other ways to immigrate (asylum, U visa) but you have not described any facts that make those seem possible.

Per sections 212(a)(9)(i) and 212 (a)(9)(ii) of the Immigration and Nationality Act, persons who have accumulated one year or more of unlawful presence (after 04/01/1997), and then left the United States [YOUR SITUATION], cannot return to the U.S. for 10 years. A waiver of the ten-year bar is available only where extreme hardship to an APPLICANT’S CITIZEN / PERMANENT RESIDENT SPOUSE / PARENT can be established. Hardship to the immigrant himself or to the immigrant’s children is not a factor (even if the children are U.S. citizens). Feel free to contact an immigration attorney for a in-depth consultation – including how to come to U.S. when your children turn 21.