I think you're very wise not to live with your partner when there is this degree of hostility from a SKID.   I think it was a mistake on your part to write a letter to SD saying you're sorry for something you didn't do, but that she lied about.  What I would do in your place is make absolutely zero effort to establish any sort of communication with this obnoxious girl.   Your partner isn't exactly helping, either, is he?  "Tried talking to her" doesn't really cut it.  Every time she is rude or sullen with you he should have come down on her head like the wrath of goddess.   Going forward, I would pretend the girl doesn't exist.  Say hello when she comes into the house, but then nothing.  No polite enquiries about her or what she has been doing, just nothing. No birthday presents or Xmas presents, if you have been in the habit of getting these.  The most I do for my two (now adult) SDs is sign the card that DH buys.  Do not cook her any meals - if you all eat together then your partner can cook.  Don't make yourself uncomfortable so that your partner can feel comfortable. I would also suggest you read "Stepmonster" by Wednesday Martin. 

Never, ever apologize to a lying, manipulative,  POS manipulative failed family progeny... no matter what. Ever. Period. Dot.

Quit accommodating her crap and adopt a zero tolerance total misery perspective with her.  The misery being delivered by you for her to live.

"I don't have the balls to lay down the law!!"

believe me when i was 3 yrs into being married and 2 kids together and told hubby i was done that we were getting a divorce because he was a pathetic husband and dad who had no balls to set his kid straight, the next time his adult ss came back from college hubby picked him up.

that lazy disrespectful piece of shit was bleaching and cleaning his mouldy bedroom and bathroom within the hour, lost the right to lock his door and his dad could walk in to check on him without knocking.

that car ride home for ss back from college wasn't a pleasant one. Its funny how much difference there is when the father grows a pair and rains down like thor on his kid!!

OK, from the top:

"she would either ignore me or just run out of the room like she hadn't heard me"

She may not be your child but you should absolutely have called her out on her rudeness. What would a teacher do in the same situation? Or any other adult. Direct rudeness gets a direct response.

"she would stand in front of me and talk over the top of me"

Nope, zero tolerance for that. "Wait your turn, child, the adults are talking." Seeing as your partner is too much of a wimp to do it. Again, what would a teacher do?

"she would pull my hand out of his so he could hold hers in stead"

And why does he tolerate this? Sitting on daddy's lap at age is kinda creepy. Next time, take a photo and show it to him later ... ask him if he'd mind if you posted it on FB. And if she tries to take his hand out of yours, hang on for grim death. If he won't put her in her place, then it's up to you to establish yourself as the alpha female. 

"I always was expected to prepare a seperate meal for her"

Don't do it any more. She eats what is made for everyone. If your partner wants to cook something else for her then let him BUT make sure he understands how insulting that is to you - not only is your food not good enough but he lets his get cold while preparing something else for his princess. If he does it again after that, just stop cooking for both of them. They don't deserve it.

"She told he mother I make no effort to talk to her, that I make up rules and yell and scream at them."

So she's a liar too... bravo to the parents.

"Thier mother would bombard my partner with messages, calling me names and insulting both of us."

There are several ways round this. One is to refuse to answer text and phone messages and to insist on email only. A clear statement of his intentions to do this must be made in advance. Then he has to follow through. OR, he insists on getting Our Family Wizard for all communication between them (it can be court-ordered). But he doesn't have to sit there and take her abuse.

"he assured me he was going to do better with the kids."

And does he? Or is he still the same spineless wimp when faced with his precious princess? What progress has he made? 

"He father has tried talking to her"

And that helps how? What about discipline? What are the consequences for her appalling behaviour? Does he realize what kind of a monster his non-parenting is producing? 

"I wrote her a letter saying I was sorry"

I think the other posters have been quite clear about what a monumental error this was.

Stop giving the brat any kind of power. Treat her like you would any other kid.  If she wants to park her bum in her bedroom 24/7 then let her. If she wears her "angry face" all the time, maybe tell her what my parents used to say "if the wind changes when you're wearing that face, it will stick and you'll be stuck wearing it for the rest of your life". When she sulks, kill her with kindness.  Oh, and tell your partner that it's time he started actually parenting her rather than pandering to, and coddling her. Remind him that the purpose of parenting is to raise a decent human being, not to make them think they're the kings/queens of the world and better than everyone else. 

So, your partner has raised a disrespectful little brat and gives her no consequences.

Just because he only sees them EOWE doesn't mean he shouldn't be their parent and not their pal.

If I was having a conversation with my partner she would stand in front of me and talk over the top of me. If I was holding his hand she would pull my hand out of his so he could hold hers in stead. when we were sitting on the couch together holding hands she would sit on his lap and try to push my hand out of his, often scratching me in the process.

And your partner did NOTHING when these things went on?

If he's not going to give her consequences, I would be giving him consequences. And I would absolutely be calling her out, myself. If he won't demand that she have respect for you, as an adult, then you should do it, yourself. If he has a problem with that, he obviously doesn't have any respect for you, either.

Maybe breaking up with him was the right thing to do. If you peruse this site, you will see that when these kinds of SDs turn into adults, it just gets worse.

You've put up with this crappy treatment for 8 years, you've set the pattern you are okay with being second rate to his special snowflake. He has TOLD you he is unwilling to address the issue. Personally, I would move on. Things are very unlikely to change at this point. 

Don't feel any guilt. You've done nothing wrong. SD seems like a miserable person and always will be.

OP, your man has done very little to address the situation in 7 years. 

So he's unlikely to do much more from now on. I think you need to define to yourself what sort of a relationship you want. As other posters have said, you could completely disengage from the brat, stay in your own place and just date him. But don't hold out any hope that when she reaches 18, 21, whatever, that things will magically improve and she'll launch and be independent.

Because she very likely WON'T. 

If you want a relationship with a view to living together in peace and harmony then I don't think this is the man for you.

Also, it's actually a physical turn off to see a grown man cowed by his daughter/mini wife. Let me tell you that from experience. 

A spineless guy who can't find his balls and lets his Princess walk all over him, rule the roost and call all the shots, is NOT sexy at all.

OMG you are giving me a glimpse of my life. Though I've backed her out of my life I know the issue is still there I just don't have to watch the shit show anymore ahhh this makes me feel so much better. 

Screw POS fathers and their bitches of daughters.