Two hall lights not working with light switches
It sounds like a 3 way switch setup, use caution when working with 3 ways because there can be power and the light is off (no return path)and you can get shocked by providing a path to ground.
  I would start with the 2 loose white wires that are twisted connecting them to the new fixture black and white. then checking positions of switches that is the most likely solution.
I have seen 3 ways wired dozens of different ways and this is one.