I think you, as the stepmom, probably need to back off a little bit. Yes, it's unfair that she has this animosity towards you, but she's 13. 10 months is not a lot of time for her to adapt to the new situation. Remember that she is a child and is suffering for something over which she has no control (her parents being separated and you being a permanent fixture). I don't think she should be rushed off to a shrink for acting like a 13yo in a tough situation.
I am categorically opposed to forcing visitation on a child. There have been a couple of occasions where my dss (11) has not wanted to come for the weekend. These instances are handled on a case-by-case basis - usually resulting in Ryan making the final decision. If he doesn't get to make the final decision, he gets to know why.
All that being said, you deserve to be respected in your home. I don't mean that she has to speak to you if she doesn't want to - I mean she shouldn't be allowed to be rude or mouthy to you, and if it really bothers you, she shouldn't be allowed to use your computer to say nasty things about you.
My stepkids know that they are allowed to be angry with me and/or their dad. They are not, however, allowed to be rude or mean. They're not required to love me and I make it clear that I know they will ALWAYS love Mommy more. This was hard for me to accept in the beginning - but since I have and quit trying to force MY IMAGE of a "blended family," life has gotten much more pleasant and my relationship with the kids has improved dramatically.
From what you describe, I see a 13yo being surly and noncommunicative and bad-mouthing her parents. I don't think one bit of that is abnormal. To me, the worst part of this whole deal is that this appears to be affecting her relationship with her dad. How does she relate to your Dh when you're around? How about on Tuesday's when you're not around?
I can see several explanations for her behaviour. One of these is that she feels threatened by you and your children (possibly especially the new baby). She needs to know unequivocally that her father still loves her. The best way to do this is for you to back off a little bit and allow her and her dad to spend A LOT of quality time together.
This is one of those times that "Stepmom" totally sucks. No one will give you any credit, or appreciate your efforts, but no matter how unfair it is to you, you have do what's best for the girl. Because "step" notwithstanding, you are the mom. And that's what moms do.