I think the OP has been given far too hard a time tbh. As well as the dissection and slating of her relationship, which is unwarranted, there's the criticism of her parenting. Lots of posters banging on about how leaving (or that much more emotive word, abandoning) your child makes you a shit parent. So what exactly are parents meant to do when a relationship breaks down? Stay together for the sake of the children? 
. Because the alternative is that one parent has to leave. And frankly, unless that parent does so without a backward glance, or any subsequent attempt at contact, I don't think they have abandoned their child, or that it should be referred to in those terms.
I would feel very sorry for any mother or father NRP reading this thread and all the comments about what shit parents they are for leaving a child.....
I also find all the brouhaha over payment or otherwise of maintenance, and being drunk in charge of children faintly ridiculous. WRT maintenance, not everyone is entitled to maintenance, I earn 10x what my Ex earns declares so am not entitled, nor have received, a penny. I've never used that as a reason for him not to see our DC, and wouldn't.
OP, I do agree that your priority should be re-establishing your own relationship with your DD, and keeping on with the things you already do. I do agree at this stage contact should be just the two of you, not your DP as well. You also need to reassure your DD that you're not trying to take her away from her dad, but also how much you enjoy spending time with her etc.
As to your DP, he needs to find a way to get to the fortnightly contact centre appts. I think you said this goes back to court in June - it would be good for him if he can show 3 months of proper contact between then and now, proof to the court (hopefully) that he is getting his act together.
In between contact, to some extent you're at the mercy of his Ex. She may be entirely reasonable, read his letters/emails to the DC, and it's they who are not interested in replying OR she could chuck them straight in the bin/delete them. There's no way of knowing, but your Ex has to try. If as you say he is depressed, there is a tendency to think whats the point. But perseverence is the key. Many of the suggestions made have been good ones, little daily letter or emails, could even just be pictures or cartoons, things that will entertain them and make them laugh, things they like, or have done together in the past. Children aren't good at talking on the phone at that age - but they will know and remember if their dad does (or doesnt) call them every day. Which he needs to - even if it's for 30 seconds.