How do I complete little girl quest after smashing red jewel brooch?

It's OK not to give her Red Jeweled Brooch.

At this point you have three possible ways to continue her quest:

  1. Visit Iosefka again and you will be able to send little girl to her clinic.
  2. Visit Oedon Chapel and send the girl there.
  3. Don't do anything - and the girl will leave to search her parents, and eventually something will happen to her in other location.

I've tried to minimize spoilers, but if you want to know details anyway - here is her quest progression - Viola.