I disagree with your first point. If she legally is responsible, and the fence needs replacing, and due to her plant, I'm pretty sure I have a strong case as to why she needs to pay towards sorting this.
You're "pretty sure you have a strong case"? Either you've got absolute proof of ownership, or you've not - hence my point about you're not going to get very far to force her to replace it. You would need to take her to court and accept the risk that if you lose, yours and her legal fees will cost a lot more than the fence would cost to replace.
Hence most people in your situation would just stick a new fence in front of it, and know that it's now yours.
She'll need to cut that ivy looking bush back as it'll eventually damage a new fence anyway.