I've moved my family out of a noisy flat before, as the tenant above was a nice bloke, but had a loud hifi and TV, with speakers on the floor, had regular enthusiastic and very prolonged thrusting-type sex in a squeaky bed, ran his washing machine in the middle of the night to benefit from economy 7 electricity, meaning we were woken up by spin cycles at 3am, and other things. He thought we were super fussy.
It's hard to tolerate each other in flats.
I actually think you're in the right, but you have to get on with people, so compromise is needed. The carpets are a good idea. You could go further and lay special carpet underplay with a sound insulating membrane incorporated within it. PigletJohn on MN knows all about this stuff, if you'd like to know more.
I do think 6.30 is a bit early tbh, and I wonder if you could push that on to 7am or later by putting her to bed 10 minutes later each night until her body clock has shifted? (If that fits with morning schedules, of course).
However the key to this has to be communication. Obviously your neighbour doesn't get the whole children thing, so you are going to have to get her up to your flat socially, so she sees the people behind the noise.
Best of luck.