How to determine the cause of bacterial meningitis?

[GoGoGirl]

I'm working on a homework problem in which the patient is known to have bacterial meningitis; I am trying to figure out how you could determine whether the cause of the meningitis is due to Neisseria meningitidis or Haemophilus influenzae.

I know that a lumbar puncture and bacterial culturing is used in meningitis diagnosis, but I already know that the cause of the illness is bacterial (and I'm pretty sure the CSF would display similar levels of glucose, protein, etc no matter which bacterial culprit it is). I could grow them on chocolate agar, but I'm not sure how this would help differentiate since both would be able to grow. I could do an oxidase test, but both possibilities are oxidase positive. Both are encapsulated also.

Would I culture the bacteria, do a Gram stain, then a microscopic observation to determine the morphology of the culprit (leading to a conclusion)? Whould I do a serological test?

Any help would be much appreciated.