Ingrid Helene is my first suggestion, but its not exactly a frilly, girly middle name. Maybe
Helena would make it more girly?
I don't really know how to define girly names, nms really, but I'll try my best for some other suggestions:
Ingrid Rosemarie Ingrid Charlotta Ingrid Lilia Ingrid Lorelei Ingrid Amelie (e is accented, so it sounds something like Ah-may-lie)
Ingrid Serena
Do you like any?