Hi Make, I have ummed and ahhhed about whether to post, but decided I might be useful, so I am posting. :) Just ignore me if I'm not!
I am almost 18 years down the line from where you are - my DS will be 18 this year. So, to start with the best news, one day you will look back on today, and it definitely won't eat you up and tie you in knots and make you feel sh*t, like it does right now. Instead, you will just feel regret - a sort of gentle sadness and a wish that things could have been different.
And you will have a nearly-grown-up daughter who has survived childhood without a dad. And you will have been able to give her everything she needed. Everything. :)
You've said several times that you don't understand why you feel so bad. My own answer was that it was about 'giving up the dream of a happy family'. When I got pregnant, I hoped for a 'happy ever after' - I think most of us do, since it's so ingrained in our culture - and when things went wrong, that dream died. It hurt a lot, and it made me very sad. I'd go so far as to say it was like a bereavement: I mourned the 'death' of the happy family life I'd hoped for for my child and myself. I recognised it, but it still took time, just like any bereavement takes time.
Even though I knew DS's dad was a b*stard who would never be any good for me, it took me a very, very, very long time to get wise to the fact that he would be no good as a father, either. I think this was basically because I loved DS so much that it made absolutely no logical sense to me that his dad wouldn't love him the same: DS was obviously entirely loveable, so of course his dad must love him, I thought - it must just be a matter of me 'fixing' something, or putting the right little something-or-other in place (though I didn't know what it was). That got me stuck into a long, complicated, painful and totally unsuccessful cycle: trying to arrange contact; applying for a contact order (me asking the court to make him see his son - I failed); taking my DS to see him; still taking him hundreds of miles to see him when he moved; me staying overnight in a big city so DS could stay with his dad; paying train fares (when DS was old enough to travel alone), etc...
Just to emphasise: court was not a solution for me. I applied for a contact order because I wanted DS to see his dad, and I wanted him to be able to rely on it. The court did not grant it. They told me no parent could be forced to see a child. They gave him a parental responsibility order because he asked for one (godness knows why; he's never made any kind of use of it) in the same 5 minutes that they declined the contact order and he said he was moving 150 miles away.
And do you know what? Over the years, although his dad made lots of noise about contact - and specifically about how I was stopping it because I wouldn't co-operate enough - he didn't actually make it happen. In 17 years, he has asked to see DS perhaps half a dozen times. Perhaps. :( 
When DS was about 12, and old enough to travel by himself, I stopped doing the running - and contact fizzled out. When DS was about 14, the CSA finally caught up with his dad - who has never paid any maintenance - and he left the country to avoid it. He hasn't seen DS in 4 years, and is in email contact about once a year.
The headline for you is this: if your DD's dad wants to see her, he will make it happen. If he doesn't want to see her, you can't make it happen. Nothing you can do will ever be enough. Nothing. That is painful and upsetting, but I am afraid it is true. You will just tie yourself into years of upset if you keep trying. Please don't repeat my mistake.
However, I will add this... This is the bad new, maybe... Some people are advising you to cut him off and refuse all contact. But sadly, you can't do that. If your DD's dad chooses to apply for a contact order, he will get one; if he asks for contact, you will be expected to co-operate. Cutting him off totally is not an option.
This is one of the main reasons it is so stressful and upsetting for mums like us, IMO: we are locked into some kind of long-term relationship with someone who has treated us like sh*t. With any other kind of relationship, you can walk away and put it totally behind you: but if you have children, you can't walk away. That's really hard, and it is probably one of the main reasons you are so down.
So, what you have to do is develop resilience. You have to break free from him inside your head and emotionally - while still engaging with him regarding your daughter, if and when he engages positively with you - and only then. That's hard, but it's do-able.
It is very, very important for you to find ways of changing the ways you respond to him. You can't change his behaviour, but you can change your reactions to it. You can learn to detach - nothing he says to you or about you important, unless it positively supports your DD and you. You can learn how to recognise that when he (or his mum) tries to make you feel bad, this is their problem, not yours. You can learn to refuse to engage if they are not 'playing nicely'. You can learn to work out what your 'bottom line' is, and stick to it. You can learn to focus on what your DD needs, and what you need, and ignore anything he throws at you that does not meet your needs.
All this takes time, and needs support - which is why counselling and CBT are so useful - but you can do it. Then you will join the ranks of the incredibly strong and confident women, who have survived huge stress and difficulty, and are all the better for it, in the end!
What I have found is that I have a little strand of sadness and irritation - like a thread running through a woven scarf, maybe - that I accept is part of me, and am kind to, and pay attention to, but which does not drive me at all any more. :)
Maybe think of me as a message from your future 
... You are already moving forward - you have already applied for a course, which is fantastic! You will look back on this time in years to come, and feel proud of yourself for what you went through, and how you survived, and what you have achieved. I promise you will! You have the best motivation in the world - your DD and yourself!