Housing Act 2004, Section 234
The Management of Houses in Multiple Occupation (England) Regulations 2006, regulation 3 and 7
The above regulations put the onus on the landlords/managing agents of HMOs to ensure the property is clean and well maintained.
Whilst this is not a guaranteed defence for your tenant, it would likely play a part if he can prove the fall in cleanliness standards to below satisfactory conditions.
Having said that, that does not give the tenant a right to withhold rent, and you are within your right to seek an eviction. There is a due process to be followed, and by the description of your case, they have not followed it.
It is also worth noting that Deregulation Act 2015 Section 33 protects the tenant from retaliatory action. So I would not be looking at Section 21. Instead, I would look at section 8 for non-payment.
But, there is a risk the tenant may win in part if they truly have a case under the aforementioned acts.
My advice, try and mediate and get them to pay the rent in arrears on the basis you will get a cleaner to come in regularly. They can be inexpensive and save you a-lot of hassle, especially with HMOs.
Alternatively, speak to the other tenants to corroborate the story and evict the nuisance tenant. But only if you can be sure they are a problem and the current tenant is not just trying to find a way not to pay rent for other reasons. I.e. complaining tenant would have to catch-up on unpaid rent.
Any type of eviction can drag out for years. My friend couldn’t get rid of a non-paying tenant for 18 months. The backlog is getting bigger not smaller.
Whatever route, mediation is always easier. In your shoes, I would get a cleaner and offer a 5-15% discount (depending on what you can afford) on the arrears to settle. Only take legal action if you absolutely have to. At the same time, issue notices to all tenants about the cleanliness as that could become a problem longer term, especially for the general state of repair of the property.
This is by no means legal advice and you should seek independent advice if you are unsure on how to proceed.