A good paed should help you to understand the problem with a good assessment of the situation. I have suggested some things to (re)consider and hopefully help you along with getting this sorted.
You will know better than anyone what has been tried and worked/not. But like a lot of approaches with kids sometimes it isn't what's been tried but how. So it is always worth retrying things such as those below.
The things I would want to consider initially are:
1 sleep history - has this always been the pattern, or has it changed? (Possible reasons why)
2 the child's response to the lack of sleep (are they anxious, unable to wind down,)
3 mental stimulation
4 physical stimulation
More re 3) and 4) what is the likely impact of these on sleep regulation? For example, mental stimulation - tv, computer games, other electronic games need to be minimised when trying to look at improving sleep patterns. (TV can interfere with melatonin regulation - also computer games).
Physical stimulation - presence of caffeine in the diet (chocolate, cola, etc). Also is it worth reducing refined sugars temporarily and see if this has an impact? (Wont work if there is high levels of eg tv/computer games though). Sugar/additives in sugary things could be stimulating.
Is the child getting a minimum of an hours physical activity? If so, could try increasing this and then consider the impact. (Try for period of time to get a reals sense of the potential impact eg a minimum of eg four consecutive days).
Re-establish a bedtime routine that winds the child down. No stimulants (ie no tv two hours before bed, no hot choc etc). Keep to the routine, even if you are tempted to bin it (as I have when totally exhausted with a poor sleeper). Bath, book, bed is a good one. Keep the same bed time.
Apologies if this has all been tried. I don't know the details of the situation, and I totally empathise with the frustration of having a child who sleeps poorly. And you are right, she will be much more able to manage the challenges of school with a good night sleep. Try not to get too hung up with this at the moment. Aim to increase the amount of sleep she gets over eg a week, and see how it goes. Good luck.