A., you can't watch your daughter like a hawk if you are working full time. Are you sure your daughter has not been roughly handled by her caretaker? If it hasn't crossed your mind then it needs to. Just a thought, not accusing you of anything or them so don't be offended please. I would expect the doctor to be knowledgable enough to know if it was recent or not.
I took care of a 12 month old foster child in the 80's that had hip dysplasia in both hips from birth and she had a body cast from armpits to ankles with a bar between with the diaper area cut out. I have no idea what you are looking at in regards to a cast these days for hip dysplasia of one hip but I would expect your daughter to be unhappy about it in the beginning because she will be curtailed in her moving around I would imagine. Someone else could maybe tell you better about what to expect. She will need a responsible caretaker to see to it that she is taken care of properly during this time. You can get through this A., so don't panic, there are definitely worse things as you said. You learn as you go like I did. My foster baby had been laid in the fields as her mother worked and when the doctors checked her cast and found maggots, they removed her from the home and I got her to care for. We made it through everything, even the no sleep for months from having to get up every two hours in the night to clean her diaper area because of the sores from the maggots. It was not easy, just a job that needed to be done for this little on, but so worth the time with her and watching this sweet baby begin to come out of her quiet shell and smile at me as she moved one little ankle round and round.
Good luck A., I wish you the best.