So are academy schools required to submit their admission arrangements to the Local Authority for approval, then
They are required to consult with the LA when setting their admission arrangements. And by 1st May they must send the LA a full copy of their admission arrangements for the admissions round starting that autumn. They are not required to get LA approval but at this point the LA can lodge an objection with the adjudicator.
who employs the ajudicator
The adjudicators are employed by the Office of the Schools Adjudicator which is funded by the government. Adjudicators are appointed by the Secretary of State for Education but operate independently of the DfE. They have been quite happy to take academies to task over their admission arrangements including, at times, chains that had been praised by Gove.
what powers do they have over sponsored academy chains
They can consider the admission arrangements of individual schools within the chain or, where appropriate, groups of schools or even the whole chain. If they find the arrangements do not comply with the Admissions Code and relevant law the admission authority must change the arrangements. An academy that fails to do so is in breach of its funding agreement.
when sponsored academies have previously refused to get outside help after a bad ofsted visit, they did so with impunity
A school does not have to get outside help after a poor Ofsted. They must draw up a plan for fixing the problems which gains Ofsted's approval. If they can convince Ofsted that they are able to fix their problems without outside help they will be allowed to do so.
what is the enforcement system
Decisions by the Adjudicator are binding on the admission authority and can only be challenged through judicial review. Barring such challenge decisions will be enforced by the Secretary of State if necessary. Given the state of the law and academy funding agreements the Secretary of State cannot refuse to enforce an Adjudicator decision. A school that refuses to comply risks losing its funding.
If we imagine a situation in which the decision was not enforced the school would face large numbers of successful appeals as any appeal panel would have no choice but to conclude that the admission arrangements did not comply with the Admissions Code. So even if there was no direct enforcement there would still be a very strong incentive for schools to fall in line.