Really sorry to hear this.
Will you be able to get proof from the plumber and electrician who did do the work on your kitchen that they did it and you paid them for it?
Sounds like you need to sit down and work through all your expenses relating to the house - maybe from before he moved in, before the kitchen was done, work out all the bills, the amount that they increased by when he moved in.
Point out that your understanding was that when he moved in he was going to pay half the day to day living costs of being there - because it would be cheaper for him than rent (if that latter bit is true?) but that the mortgage was never on the table.
Did you ever have a discussion with him about how much he was going to pay when he moved in - even if you had to wait so long for a first payment?
If so - you need to work out exactly what you paid and what he paid every month through the relationship, along with notes about things changing (making changes to the house, his dd moving in, differences in utilities used, having to go full time, debts you accumulated due to him etc)
For example - did he ever pay his 25% share of the council tax that would have gone up as he moved in (were you the only adult in the house beforehand? or did you pay the entire increase? Did he ever pay anything towards it even when he was notionally paying towards the 'mortgage' because you could certainly argue that if you had realised that he thought he was just paying 'mortgage' then he would have had to have paid for his half share of the council tax rather than the extra 25% that was added when it went up from one adult to two adults (if that's the case).
Unfortunately I think this is a warning shot over you - and that you have got to be uber prepared for any future meetings you have - mediation or court - so that whatever figures he produces, you have ones to fight back with.
It might also be worth thinking of some really incisive questions for the next round of mediation or court - getting them to break down for you exactly how much at each point of your relationship how much equity he could expect to get in your house for how much input from him...
So - before he moved in, helping out by fitting a kitchen, albeit at mates rates - shouldn't expect any value - he offered you a service a t a price, not a price plus increase in percentage value of your house at some future point in time of his choosing. make sure you don't include mortgage costs at any point.
First month - you incurred £xx extra charges in food, utilities, council tax, etc etc, expected payment of £yy, received nothing
Second month - you incurred £xx extra charges in food, utilities, council tax, etc etc, expected payment of £yy, received nothing
Third month - you incurred £xx extra charges in food, utilities, council tax, etc etc, expected payment of £yy, received nothing
Fourth month - you incurred £xx extra charges in food, utilities, council tax, etc etc, expected payment of £yy, received nothing
...and so on - with a running total of how much you were expecting in actual money that you were actually losing in paying for all these extra things and how much he owed you in total at the end of each month.
Then get to:
22nd month - you incurred £xx extra charges in food, utilities, council tax, etc etc, expected payment of £yy, received nothing -
By this time, guessing the amount you have paid out and that he owed you will both be substantial
23rd month - you incurred £xx extra charges in food, utilities, council tax, etc etc, expected payment of £yy, received £600 towards costs
So start to then change the figures so that his total balance goes down by £600
24th month - same again, through to the 34th month - hopefully by this time you are still showing that he is in debt to you for his share of the costs
Not sure what month(s) he did the changes for his dd - but see if you can find a way to include them to show that changes were made, but that it made no difference to the value of the house, not sure if you paid any money for them or if he paid for all of the supplies and did the work. But put in costs that you would incur when you change it back (whether or not you want to or will, when you can afford to - sounds like you can't at the moment thanks to his debts!) to offset against any of his values and to keep re-empahsizing the point that it wasn't a change you wanted to make to the house.
I think it's really important to emphasize that you don't like the changes that he caused to be made to the house (regardless of whether you do or don't want to keep them) - that they reduce the enjoyment you get when living in your home and if you had the money you would like to change it back to how it was before, that it was an easier/nicer/better house to live in then, and that if you were to sell it, it would be more sellable if the changes he made were removed. That it is certainly something that you would never have done if he hadn't come to live with you and need somewhere for his dd to stay, and that you want to redecorate so that you have a lounge and a conservatory again, rather than a teen girl bedroom and a conservatory/lounge which is one thing in the summer but not great as your only lounge to be cosy in during the winter.
Once you have all the figures, put in a line right from the first month he moved in saying something along the lines of 'value ex can claim from my house' with a couple of ?? in the running totals column. Say that you want to understand exactly at what point he thinks he should be able to claim value from your house, the reason why, and the exact amount, and how does this link to the very tangible amounts of money that he owes you and you are still in debt for and having to work extra days a week to pay off. Then get them to fill in the whole thing, month by month.
Also make sure they put in whether this is a % of the overall house value, a % of the increase in the house value since the changes were made/a particular point in time (in which case ask why he never thought to mention it to you, because you would have most definitely got a valuation done and something formally drawn up so that it would show you were both in complete agreement and there could be no misunderstandings) or just a fixed ££ value, regardless of the overall value of the house.
I think if you can work out a seemingly simple spreadsheet of values that he has to fill in that shows the amount that he is expecting to earn off you from a house that hasn't increased significantly in value, that it will show that he is expecting to take from it more than he ever puts in, and has completely unrealistic over-entitled expectations - not least because you can probe as to why he thinks should start to be entitled to £aaa rather than £bbb for the first month vs the third month - does he think that he should be entitled to a big hit from the word go, so would he then expect to get the same from you if you broke up a year earlier or 5 years later? Or why does he think that he should get a percentage but how did he work that amount out? And not sure how the £600 related to the actual mortgage payment you made but ask him about his living costs and why he thought that you would be paying all the other costs, remind him that he was always late paying which meant that he knew he wasn't playing the actual mortgage because if he was that would screw the mortgage up, he never paid into it directly and it was never discussed. And so on - also take it into the future - with what would have happened in 5/10/15 years time and when the mortgage ended - would he have expected to have overtaken your share in the house if the figures worked like that despite paying just mortgage and no living costs - or would he have stopped at 50/50 ownership - and if so what about the equity you had in the property beforehand and so on (and if you ahd equity in the property beforehand that gets counted at the end of the mortgage then why is not being taken into account now? And when there was no mortgage to pay - what would he have paid then - and why? Or would he have expected you to pay for everything? Or would you have split everything - so if so, why is he not even doing that now?
But hopefully if say he is expecting to get £1600 of your house (I don't know the values sorry, so just making it easy for the maths!) for every month he was paying you £600, thus making a profit every month, when you're not making the same amount on your own house, then how did he ever think that was fair? And why weren't his debts to you being considered at all?
I think if you do it month by month, it might at least make the mediator stop and think as well as make your ex think, to point out how crazy his claim is. Also to point out that there's no automatic rights (can anybody confirm this?) to getting part of a house as many women have been thrown out of men's houses in similar circumstances, so what makes him different?
Good luck...