The enotes version of "A Retrieved Reformation" does not contain the sentence "Jimmy Valentine knew the game was over." I have checked several copies of the story available on Google and don't find this sentence. You must be quoting from a different edition. However, "the game was over" was probably just a common saying in O. Henry's day. The idea is that Jimmy's pretense to be Ralph Spencer, an honest businessman, was totally demolished when he exposed his suitcase full of specialized safecracking tools and opened the supposedly impregnable vault in ten minutes. The game was over because there were so many witnesses present, even including his beloved fiancee Annabel. He could not resume his false persona as Ralph Spencer. Jimmy started to walk towards the front door because he believed that his relationship with Annabel and her entire family was destroyed forever. This is how the paragraph reads in the enotes version:
Jimmy Valentine put on his coat, and walked outside the railings toward the front door. As he went he thought he heard a far-away voice that he once knew call “Ralph!” But he never hesitated.
To make matters even worse, he found Ben Price waiting to arrest him before he even made it to the front door. Jimmy is feeling so desolate that this final blow makes no impression on him. He has already lost everything he valued in the world.
“Hello, Ben!” said Jimmy, still with his strange smile. “Got around at last, have you? Well, let's go. I don't know that it makes much difference, now.”
Once Ben Price makes it clear he is not going to arrest him, Jimmy still has the problem of doing a lot of explaining to Annabel and her family. This is not included in O. Henry's story, but we can assume that they will be understanding and forgiving because they will accept his word that he is completely reformed and because he made such an heroic sacrifice when he saved the little girl trapped in the bank vault.