Desperate, final attempt. Last attempt to succeed. ‘Broke” is used in the sense of succeeding or failing miserably. You may attempt a new enterprise which may end of failing and putting you in the poor house. (Compare to Pull up your stakes, Tie up loose strings or Swan song.)

“I’ve asked her several times to marry me, so now I’m, going for broke and will buy an expensive engagement ring and offer it to her one last time and  hope she accepts it.”