I've a dd at the school, so believe I can help.
The school I have my heart set on is LEH, but I can't find any information on what the entrance requirements are for 7+
Have you visited? If you have, they give out very detailed entrance guidance notes to visiting parents, so the entrance requirements are extremely clear. If you visited but didn't get them, then the school secretary is very approachable. I'm sure she'd provide them to you happily.
Like your dd, mine too went to a school which was not a natural feeder. We covered most of the requirements ourselves using Bond books, to reinforce what she had already covered. But did decide to tutor our daughter for five tutor sessions to cover the few Maths topics that she had not covered at her previous school. I wouldn't say she was intensively tutored by any means. We did all this in the Autumn term, (Oct - Nov) so in the few months before the entrance exam. Some schools, like J&J mentioned above, prepare your children for you, but my experience is that in the large part, other parents have done most of the preparation with their girls prior to the entrance, and that a year of tutoring is simply not necessary.
The exam itself is quite relaxed, not like an exam at all. The girls go to a classroom, and do some Maths & English. They can put their hands up and ask a question. They go out to play and have a chocolate biscuit. It's more like spending a morning at a new school, but your work gets marked at the end of it. There's also an interview, girls seen in small groups where they read an unseen text to the head and discuss the passage.
It has a reputation for being competitive and extremely academic. I would say that if your child is a good, natural fit, she won't actually notice that at all. It'd just be a great school to her. Sports are very important. There is proper sport every day so if that would make your child unhappy, you'd need to consider that. I have met girls who just hate the sport!
Most children are offered a place in the senior department, though it is not guaranteed. I believe each year there is a handful of parents who are advised that it may not be the best route for their daughters and this process starts very early, there is no sudden boot out at the end of year 6.
Hope that helps.