Based on my personal experience as a step-daughter and the wife of a step-father, I know you're not in an easy position. Especially as the "new" wife and recent step-mom to teens. And all the more if the ex-wife does not speak well of you. It sounds like you're the unfortunate target of a whole lot of feelings, many of which are probably lingering from the divorce and the split of the original family. If you were in any way involved in that split, then the feelings would be amplified all the more.
A quick side note about the lying; your SD does not want you meddling in her private affairs. Privacy toward parents is completely normal in this age group, and there isn't going to be a realistic way for you to get yourself invited into this girl's life.
So, I would stop trying. You and your husband would do well to decide what the groundrules are for your blended home. This should include everyone treating each other with respect, and everybody doing their fair share to keep the household functioning smoothly. And your husband will need to be the enforcer – to whatever degree he succeeds, you'll have some basic courtesy and cooperation. This won't be an easy task, though, unless he's already on good and respected terms with his children.
For yourself, I'd suggest you just respect yourself, and behave with consistently kind but cool dignity around your SD. Take care of your own needs, and let her little cruelties roll off of you. Unless you were a home-wrecker (and she's likely to see you this way even if you had nothing to do with the divorce), she won't like you no matter what you do, BUT she may come to respect you if she learns she can't "hook" your feelings. Learn to take a lot of deep breaths, and take occasional time-outs if you feel she's getting to you. But make it look like you just need to go do something important.
This girl does not know enough about you to really hate you, so try not to take it personally. She hates the situation, the divorce, the torn family. She may be extremely angry at her dad for all of that, and you are a handy target. She's young, and hasn't had much life experience. And she's a teen – a trying enough time for most kids even without the domestic complications. If you are able to allow her to have her feelings without resenting her for it, life will be somewhat easier for all of you. Resentment doesn't heal anything.
My best to you, D.. The next year or two will probably feel very long. But deal with your own feelings – that will give you plenty of work. You can grow in the face of all this hostility. Keep picturing yourself as a calm, determined woman, a happy wife, a dignified human being. And then be those things. If you'd like any additional support in this endeavor, I'd like to suggest you check out a website called TheWork.org.