My girls have full heads of long, thick hair. My oldest started showering by herself only last year when she was 9. Before that, she could do her body alone, but needed help with the hair. My youngest is 8 and SHE started showering by herself last year when she was 7. Before that age, I still had to supervise to make sure they cleaned all the nooks and crannies. And I have them shower at night after swimming. Trying to squeeze in a shower in the AM would be way too much hassle (I'm kinda impressed you're pulling it off with your 6yr old!).
They've been brushing their own teeth pretty well since they were 4 or 5 (and according to the dentist, they're doing a fantastic job, so yay!).
Hair is another story - they brush it themselves, then I run the brush through to make sure they got all the knots out (usually, they have NOT). If they want a certain style (ponytail, braids, etc) I help them with that, although the 10yr old is now starting to experiment with hairstyles on her own. And both of them are perfectly happy to throw in a hairband to keep the hair out of their faces.
So I would say that 6 is a perfectly acceptable age to start grooming herself under your supervision, so that by the time she's 8 or 9, she's able to do it all on her own. In our household, we're now at the point where I just say "Ok, go take your shower!" and they do. I haven't "given" any of the kids a bath in ages and ages. It's kinda like when all your kids are finally out of diapers and you realize "they're growing up...awesome, that's good...right?...sniff..."