Congratulations Knjo - what a great position to be in. Your DD is obviously very bright, so wherever she will go, she will do really well. That is the bottom line.
Finances are obviously key here, and you cannot argue with a free outstanding state school. but stripping that out, as only you can decide that for yourself and your family, the question is where do you think she will be happiest, and get the most out of her school days.
How far is her independent school from home vs LMS, and what will the journey be like?
Academically, LMS is very strong, and discounting for the fact that it has a mixed intact, the results are comparable to an independent school I think - especially at A level as it does effectively select, requiring As in GCSE in order to go on to A levels. The school has an excellent reputation and the girls are lovely and it has a great ethos.
Sports can be a bit tricky, but I do not know your independent school. I know DD of friends going there have been advised not drop their co-curicular sports wise.
Bear also in mind that sciences are not taught separately if that is important to you. Also I have a friend there that has to choose between geography and history at GSCE as they cannot offer both for whatever reason, but that could be specific to this year, rather than a general theme.
Does your DD have a preference? What was your gut feeling during the open days? Whatever happens, you have very good choices, and the outcome of whatever you choose should be a good one. Well done, and however agonising it feels to choose, try to feel good about the decision :-)