Thinking about this a little more, I wonder whether there is an underlying problem. I know it's not your job to deal with it, but if you are going down the compassionate route, hearing another viewpoint might help.
You said that they used to be a model tenant with a good job, and now they have no job, are on benefits, and have stopped looking after the place. Have they lost their job and got stuck in a cycle of depression, not able to get another job because they don't feel they are worth it and not able to look after the place any more because it's not worth it? My perception is that is sometimes how homelessness starts. I could be completely wrong, and feel free to ignore me.
Lootman wrote:So the vast majority of tenants do not take such things personally nor are they difficult about it.
I do disagree with this. An eviction is a big thing - it affects you massively and you have to move house. That's up there among the most stressful things that can happen in your lifetime. Logically, it is not personal, but in the evenings, when people are tired, it really is personal. If there are other stresses in their lives, then it's because the whole world is against them and they are worthless. But as a landlord, you are not going to hear that as they don't want to make you any angrier with them, and make things any worse. For the vast majority, they won't be difficult about it because the landlord/tenant relationship has already broken down, and they don't want to live there any more. Look at GoSeigen's words. If you have never been evicted (or had a close friend evicted), you will never know how it feels. Even when my close friend was evicted, I was able to escape the feelings after I went home, but she wasn't, so I don't truly know either.
But back to Mike4's problem. If the tenant has already accepted that the place needs a major refurb, is there any way it can be done while they are there? If they went to stay with friends for a month, and took a few holidays (maybe not possible on benefits) during the majorly disruptive bits, can you work around them? It's not ideal, takes longer, and needs more planning, but I moved into a fixer-upper, and got people in to do all the work while living here. That would come with a rent increase to nearer market value and/or a rent holiday if they could move out for a bit. And then ask how you can help the tenant keep the place nice. Would more frequent landlord inspections help?
But as I said at the outset, it's not your job to make the world a nicer place for everyone. If you are going down the legal route, you need to make sure that the Gas Safety certificate is up to date (I don't think that's a problem!), they have had all the required information for the deposit (not sure what the rules on that were 13 years ago, but you may have already done that) and you have given them a copy of the How to Rent booklet at the start of the tenancy (which didn't exist 13 years ago, so give it to them now if you haven't already).
https://www.gov.uk/evicting-tenants/section-21-and-section-8-notices seems to cover it all.