You might consider casting your eye wider than London. Many people commute into the city by train from the surrounding counties. This would give you a wider range of schools and many have less pressure on numbers.
For non-SEN private schools you are looking at going through their normal admissions process and giving them the information you have on your daughter's needs so that they can decide if they can provide something you are happy with. However, I don't know anywhere mainstream that offers what you currently have.
School counsellors are a standard part of the American school system, but they aren't over here. Many schools don't have any and when they do, the counsellor (NHS in the state sector, private in the independent sector) will visit for appointments with specific pupils because of a clinical need identified by a doctor, usually after a referral through CAMHS. In other schools children would be taken out of school for this appointments. Ditto with speech therapists etc. It depends where you live and what the NHS offer locally.
Schools offer one-to-one and small group sessions with the SEN department. These staff offer support in a variety of areas depending on need - literacy, numeracy, executive functions... some schools might get a play therapist in once a week if there is demand. In the private sector all this is paid for by parents as the staff are peripatetic. In a specialist school provision is more likely to be included in the school fees. In the state sector your child would have to qualify for the provision, but it is free if they do. This would require an EHCP.
The SENCo (SEN coordinator) oversees the one-to-one and small group SEN provision and also creates a report on each child's needs which is shared with their teachers, so that they know what reasonable adjustments are expected.