If the LA won't come and do a post-adoption support assessment, which they are required to do to identify the family's need and make a request for funding from the National Adoption Support Fund, the family can contact the National Adoption Support Fund directly - their helpline is really good, and although the LA will have to be involved to assess, the Support Fund people can help to make that happen.
The Fund can only be used for therapeutic activities that will directly support the child (play therapy, family therapy, etc) but funding that can ease other pressures on the family. Applying for DLA can help, if the child/children have additional needs that mean they need far more support than other children their age? That too can ease pressure on the parents, because they are not trying to be a 24 parent and somehow work too, especially if they have very disrupted nights.
Some LA's will offer an adoption allowance, usually fixed term, and usually not very big, to help tide over a family under particular strain - these can be hard to get, but worth asking about it via the SWs in the post-adoption support team. They are extremely unlikely to be positive about the idea of having another adult in the house (you mention an au pair) because the presence of a stranger in the home has a good chance of increasing the stress/challenging behaviour in the child. However, they may be able to provide contact details of specialist childcare in the area - our LA has a co-ordinator who keeps a register of childminders etc able to care for children with additional needs of all kinds.
I hope you and your family are able to find support and reach a steadier place together