I am no expert by any means but it sounds to me like you have a child who is gifted plus learning delayed in some manner and quite possibly does have ADD - but being gifted (and female) is not going to tick the boxes to be neatly and easily diagnosed with anything. In your shoes I would be researching who the local experts are in dealing with both giftedness AND whatever else you think is going on and take her to them, not just school assessments.
I just recently attended a fantastic 2E conference and the paed who gave the keynote speech was very clear that you often CAN'T diagnose any child, but particularly a gifted child by doing standard tests in a boring room with nothing to distract them. He told the tale of a girl who early in his career he had sworn was perfectly normal. The parents begged him to come observe her at school, so he did. He could hear a commotion as the principle lead him down the corridor and when he entered the classroom this "perfectly normal girl with no problems" was standing on her desk - naked and flinging her clothes around the room. He now still does a battery of testing but diagnoses with significant weight on thorough history and QEEG brain scans.
My point is two fold - there are some experts out there who do know about how giftedness and LDs intersect. AND there are also professionals like the intern and therapist you have seen, who while sitting face to face with your DD having a conversation that is fun for them both will not see any problems at all - which does not necessarily mean that there are no issues to be seen, only that they are not obvious in that scenario. I certainly know that my own DD is SO much better 1:1 that even her special ed teacher was not aware of the full scope of her issues until she hit small group support in yr3, at which point we were asked to have further evaluations done.
