Well, that was an interesting 24 hours! Thank you for al the support, I was reading, but held off replying until things had settled.
Poor DP is emotionally exhausted, but has been brilliant 
DSD and her mum had a heated argument yesterday morning, lots of shouting and tears from both sides - culminating in DSD being hit by her mum. DSD left the house immediately, her mum slammed the door behind her and screamed loudly enough for it to be heard in the street 
DSD called and texted DP, and they spend time together yesterday before DSD went home. during the day, DSD mum tried repeatedly to get hold of her but DSD didn't reply - and told DSD, via answerphone, how disappointed she was that DSD had made contact with her Dad 
DSD and DP had a very open and frank discussion, and they have agreed that they will begin to get to know each other again, through short, regular meet-ups. DSD will be coming to our home after school for a couple of hours next week.
DP spoke to his ex, and his ex was at pains to find out if 1) he knew she had hit DSD, and 2) what he was going to do about it. His ex downplayed ,but didn't deny, hitting her, and DP gave her no assurances or otherwise as to what he was going to do. He slept on that until this morning.
Neither DP or I beleive that DSD is a risk of serious physical abuse. If we did, then DP would have prevented DSD going home yesterday. However, it is obviously of concern that this has happened and DP wants some assurances that his ex is seeking support to parent DSD - who she described to both DP and DSD herself as unpleasant, and someone she didn't know or like.
It is clear that DSD has huge unresolved issues about the breakdown of her parents marriage over 3 years ago. She knows the grounds for divorce (her mums admission of an affair), and the fact that DP wanted to try and work things out, but that her mum refused. A lot of that DSD has been told directly by her Mum, and any gaps have been filled by DSD snooping through her Mums filing, which she admitted to DP. She said to DP that she can't trust her mum, and that every time she rows with her mum, she wants to yell at her because she is angry about what she did that split up the family 
DP is reassured that things have calmed down, for now, and that DSD and DSS are OK.
As for me, I'm very conflicted about DSD being a part of our lives again, and I know it's very early days. DSD did admit that "she doesn't like me" to DP yesterday - which is a huge achievement for her. When gently pushed, she couldn't give a reason other than "she hasn't warmed to me". I think the time has come for me to seek some independent support for my feelings - so I can support DP/DSD and not impede their reconciliation..
DP has made some calls this morning and has arranged a meeting with DSD school as soon as possible - apparently, DSD has already taken one GCSE and will be sitting another soon - something DP had no idea about. Obviously, the turmoil in her emotions is not going to help her academically.