Batman ends up (legally) adopting the baby girl he saves as his daughter, and as she gets older, he starts training her in various forms of combat and self-defense, such as martial arts and kickboxing, and maybe even has her become the next Robin or Batgirl.
I imagine that before adopting the baby-girl as his daughter, Bruce/Batman would do his best to locate any living relatives who might be able to take care of her (including trying to find out who the baby's biological father is).Once he's able to legally adopt the baby as his daughter, Bruce would, for sure, make "Martha" her middle name as a way to honor his mother, but I'm not sure about what he'd do for a first name.
Also, I feel that along with having Alfred around to help him raise his daughter (who Alfred would probably see more as his daughter or granddaughter), Bruce would hire a woman to come and work for him as a housekeeper/nanny, but also having her to act as motherly-figure/female role model for his daughter.
