Your options include marrying, having her return home, then applying for her to come back with a spouse visa; having her return home, marrying her overseas, then similarly applying for a spouse visa; having her return home then applying for a fiancée visa; or marrying here and seeking to adjust her status on the basis of marriage. If you pick that last approach and she overstays, she would not want to leave the U.S. while her petition for adjustment of status is pending.
The big question, though, is whether her J1 carries a two year foreign residence requirement. Marriage won't change that requirement.

