It can be argued that Jimmy surrenders because he believes that his identity has already been exposed, and therefore, it would be pointless to run away.
At the beginning of the story, Jimmy is released from prison.
Immediately, he returns to his life of crime. A week after his release, a safe is broken into in Richmond, Indiana, and eight hundred dollars is stolen. Two weeks later, fifteen hundred dollars is taken from a safe in Logansport. Then, five thousand dollars is taken from a safe in Jefferson City. The author hints that Jimmy is responsible for all of these thefts.
Later in the story, Jimmy falls in love with a woman named Annabel Adams. Because of his newfound happiness, Jimmy decides to leave his criminal past behind. He makes plans to give the tools of his trade away to an acquaintance. However, on this fateful day, two children are locked in a bank vault. So, Jimmy is faced with a terrible choice: save the children and expose himself to Ben Price (the investigator), or leave his future nieces (by marriage) to die.
In the end, Jimmy decides to save the girls. He also decides to give himself up to Ben Price after the rescue. You will recall that prior to the incident, Jimmy had already decided to reject the criminal lifestyle. The author hints that Jimmy's goal is to live an honest life, the kind befitting a family man. So, he surrenders to Ben Price because he believes that it is the right thing to do.
Another reason Jimmy surrenders is because he believes that running away would be pointless. His identity has already been revealed to Ben Price. Furthermore, Jimmy may be tired of his old life, one of constantly being on the alert against law enforcement. So, there are quite a few reasons Jimmy may have decided to surrender to Ben Price.