Both ds1 and ds2 went to selective grammar schools in Essex. With ds1, we didn't get him tutoring - my thought was that if he couldn't get into the school without being tutored, he would struggle once there. Then, a fortnight before the exam, we gave him a maths paper to do, and he scored 40% - and dh panicked. Ds1 got lots of maths for the next fortnight - and he passed the 11+ (to be honest, I suspect that was despite dh's tutoring, not because of it).
With ds2, we got him tutored - and again, I don't think he really needed it, but it made the whole process a lot less stressful.
Ds3 did start the tutoring, but had a total melt-down because he wasn't keeping up with the work he had to do for his tutor, so we let him drop it, and accepted that he didn't want to do the 11+. In the end, he did decide to do it (I think we must have kept his name on the list in case he changed his mind), but didn't pass high enough up the list to get a place - and tbh, he was happy with this, as most of his friends were going to the local comprehensive.
In the end, we moved before he finished year 6, and we chose our new house based on its proximity to a really good comprehensive school - the ideal that Elibean talks of - that is inclusive, works hard with all its pupils, and helps them all achieve their best - and I believe that that school was the best for all three of the dses.
Ds1 is reading law, and got an unconditional offer based on his Highers (a scottish exam somewhere between AS and A levels, I believe), and ds2 got 4 unconditional offers to read advanced Maths, and has just finished Freshers Week at Edinburgh - and I am sure that a lot of the credit for this lies with the school and their teachers. But it is ds3 who has benefited the most, I believe.
I think parents are caught in a very difficult dilemma if they live in a grammar school area. The grammar schools have excellent reputations for academic, sporting and artistic achievement, and in my experience often have truly excellent facilities and staff, plus good discipline (so teachers can teach rather than having to do crowd control), and as parents, we all want the best for our children. But both the selective grammars that my boys went to (they didn't go to the same one - oh no, that would have made my life far too easy
) were very hothouse atmospheres, and that doesn't suit everyone. And if a child has been top of their class all their primary life (as ds1 was) and finds themselves in the bottom third of their secondary school year (as he did), this could be quite disheartening (luckily ds1 is pretty oblivious to stuff like this, and it didn't bother him).
Having been raised by a very socialist father, I did struggle with the idea of a two-tier system, and whether I should be perpetuating it by using it for my dses - but I wanted the best education for them, so they went to the grammars.
To the OP I would say, let your dd visit all the schools that she has to choose from, and let her see what they all have to offer. And you can tell her from me, that the girls who went to the partner grammar schools of the ones my dses went to (it was all single sex where we were), were the polar opposite of ugly and geeky. And my friend's dd, who started at the grammar school the same time as ds2, had a wonderful time, made lovely friends, is a fabulous dancer, and is about to start an art foundation course, so she certainly didn't find it to be all about the academics.