Coming at this from a point of view of a parent of a child with aspergers (bear with me, I'm not saying your dd does but the tricks learnt can be useful for all at times), sometimes it's covering all the little steps that makes the difference.
So, she has a hole in her tooth, which means the food is touching the nerve inside her tooth, and that is what is hurting (maybe show a diagram of a tooth, and show how with a hole it cant stop to food getting inside it). If it was a hole in the skin, we would put a plaster on it, but plasters dont stick on teeth (make her laugh, try putting a plaster on your tooth and see it fall off 
) so the dentist has to use something special like clay or playdough to fill the hole instead of a plaster (make sure you emphasise that it a special mixture only dentists have, and playdough/clay tastes very yucky and will not work because it will stay soft and hurt her tooth more). Explain that because the hole is there and it is hurting the dentist will make her mouth numb, like he did this time, just like 'magic cream' helps your knee/arm/hand stop hurting after you fall over (if you use 'magic cream') or numbs her arm if she has had a blood test/when she was sedated before (if it happened and she remembers it), however just like plasters dont work in the mouth, magic cream tastes very very yucky, will make her poorly and will not stop her tooth hurting, the dentist uses something special.
Agree that the noise is scary, and see if listening to music on headphones is something she would like to try to make the noise a little less scary. The suction is necessary because just like a plaster the mouth needs to be dry when they put the filling in, also if the drill is used it sucks away any tiny little bits of bone fragment created (you dont need to explain that bit to her). Not sure why the drilling, I believe it is to make the hole smooth enough for the filling but it is something I have never actually asked.
As suggested above, offer her something to hold that she can drop or use to make a noise if it is hurting and then the dentist can give her some more painkiller, but explain that she may feel the dentist touching her tooth and not feel any pain with it (much like if you have ever had stitches, or a c-section, you may well have felt some tugging and movement, just pain free)
Think I have covered the process, if not hopefully you can see what I mean. Some of the common sense stops might be able to be left out, you know your dd, but from experience I cover exactly why you cannot use clay, but only the stuff the dentist uses and so on. I do hope it helps.