OP, some professional advice. I deal with this kind of issue routinely.
First off, noise disputes can be extremely devisive. Long-term low level disturbance can have severe effects on some people, so I empathise with your situation. However, your circumstances put you at the bottom/low end of the scale in terms of anti-social behaviour.
For example, if you were to contact the local Environmental Health team, they'd ask for noise diaries, make out-of-hours visits and install noise recording equipment in much the same way that your housing association have done. The legislation they work with, in particular the Environmental Protection Act 1990, is quite specific about different forms of noise. General household noise such as you've described is particularly difficult to deal with from the point of view of the authorities, as there is no legislation that specifically proscribes it - there is an assumption that normal household sounds are just that.
If you have not already been in contact with your local council EHO, I suggest you do this. If there are other associated issues with the neighbours, in particular anything that might be described as Anti Social Behaviour, then it is well worth bringing that up, as it can be far easier to approach the issue from that angle. My local councillors are very good at getting stuck in with noise complaints, so it is worth approaching yours to see if they'll get involved. My local ASB team are also quite proactive, so it might be worth approaching your local team as well. In essence, if you're not having luck with the HA, try a few different routes. The more you kick up a stink, the more people will have to take notice. If the noise is affecting you/others mentally or physically through stress or lack of sleep, go to your GP.
Do NOT take matters in to your own hands - it's the sure-fire way to switch instantly from victim to offender, and then you are ****ed.
It would seem that as others have mentioned, your partition wall is of a very poor standard in terms of noise insulation, and you/your family are particularly sensitive to the late night noise that your neighbours generate. I live in a five year old terrace, and when the kids next door run up and down the stairs, it sounds like a piano is being pushed down them, so I can appreciate how annoying the situation is.
In short, speak to EHO if you haven't already; escalate the matter with your HA if you feel you aren't being taken seriously; contact your local councillors. Comply with any of their requests, engage with them, keep the noise diaries running. Get yourself/others to your GPs if you're that badly affected.
Best of luck.
