The age doesn't sound like the issue - 9 is quite mature in many respects and I have students much younger who can deal easily with this. It is timing and approach - she has only been learning piano for 3 months, I think you said!
On the written side, I tend to seperate key signatures as something to learn and intervlas as another. I approach a scale as a group of 8 notes. Intervals are easy - just count up the number of lines and spaces. Finding the semitones is easy in a major scale if you break it into two groups of 4 notes.
Playing them on the piano is another matter and I would use a visual approach to the balck and white key pattern rather than teaching the theory. However, I doubt a student after 3 months who seems to be struggling is ready for 8 note scales. If you want to, play C adn B major. It's not so easy to pass thumbs under, though. Better to establish a good hand position.
Anyway, that's my opinion. Just keep it simple and try to remember what graded you were at before you could make sense of all this stuff - slightly past intermediate for me!