Suing a landlord for damage to your mental health is absolutely possible, but extremely difficult. It can't be done in a small claims court.
You would need a formal psychiatric diagnosis, then you would need to establish with evidence that your condition was a direct result of the landlord's negligence/behavior. You would also need to establish that the impact on your life/work/health was significant. Something that has simply made you feel depressed/stressed would not meet the threshold. Something that caused severe PTSD that prevented you from working might.
Your landlord also has a duty to provide a safe and livable property. Having no washing facilities for an extended period is clearly not livable and the cooker sounds potentially unsafe. These would both constitute a breach of duty if left unaddressed.
You may also have a claim if - for example - you had to pay for a gym membership to shower, or eat only takeaway food due to a lack of cooking facilities, or if the faulty cooker injured you.
You say they provided incorrect information about the size/condition of the property. If they claimed a room was 'spacious' and you disagree, that's a matter of opinion. The assumption would be that you were happy with the property when you signed for it - doing so without a viewing is unfortunately on you. If you arrived at the property to find it materially different (eg, a flat where a house was advertised) you may have a case to challenge the tenancy under consumer protection laws. This should have been dealt with immediately though, you'd be unlikely to have a case if significant time has passed.
While it all sounds stressful and inconvenient, proving that the above were directly and solely responsible for a mental health condition (I am assuming depression/anxiety?) that significantly impacts your life would be difficult.
As you say, there are other factors at play - you lost a (presumably significant) amount of money when a client went bust. The neighbor dispute. You have general issues with the property being an unsuitable size, and the speed of the internet. None of these are your landlord's responsibility, and could be argued to be significantly contributing to your mental health issues.
If the internet doesn't work at all, they have a responsibility to get it fixed or reduce your rent. If it's a speed issue, not so much. They should have called the service provider to investigate though. The provider may have a minimum speed commitment that they're not meeting.
They cannot lie to you if you ask about previous neighbor disputes, however they are not required to inform a potential tenant directly in most cases. This is your issue to resolve with the neighbor/the police.
They are not responsible for your failed test or failure to keep on top of your personal affairs.
You say that the agency has offered a £300 settlement - is this in relation to their level of service when dealing with your concerns, or a settlement on behalf of the landlord for the issues with the property? If it's the former, take it. It doesn't prevent you from taking separate action against the landlord for the alleged breaches/personal injury.
A letter before action from a solicitor to the landlord setting out your issues and intent to sue if resolutions/compensation are not forthcoming would be a good place to start.
Are you still in the property?