See, I'm not agreeing with the whole "she deserves to know her real dad, right?" thing.
He's a violent druggy loser. He should never see her unsupervised, as you've said, and you've never prevented him from seeing her. So, you're doing it all right. When she's older, and hopefully by then equipped to deal emotionally with the fact that her dad is a twat, then she/you can chat and see what can be sorted out, if she wants it and he's willing.
Right now your job is to protect her from fools like this. Don't get so caught up in the must-see-her-dad thing. She needs YOU, the stable parent, to love her and keep her secure.
My ex hasn't bothered with the dc for over three years now. Like your ex, he seemed to think it was my job to facilitate meetings. It is not. He is a violent abusive drunken fool with a criminal record, and he stole from us so would never set foot in my house again.
I told him back then, and I'd say the same now-if he's interested in the children, he'd have to go through the courts. I think you should do the same.
A very wise family lawyer I know told me at the time, when I was still worrying about keeping them in contact because "he's their dad", said to me "you've done all you can. You are not obliged to keep up contact just because the other adult can't be bothered". He was so so right.
I stopped running around enabling my ex's shit and surprise surprise, he never ever bothered with his kids again.
Which, as it happens, is the absolute best ever thing he could have done for them as he is a dangerous bell end.
They are well, happy and gorgeous. They know he's out there, but they're not bothered in the slightest. Currently 10 and 7, if they want to get In touch with him they can of course. By then I hope they will know just how to deal with sociopaths like him!