Oh just stop harrassing the OP! There's a strong element of a vicious mob mentality in many of the posts I read over the last 10 or so pages, these posters should be ashamed of themselves.
Everyone seems to have forgotten that a year ago the owner of the fence agreed to share the costs of a new fence with the OP. And then she went back on her word. Her explanation was wrong - she claimed that the fence wasn't hers because the posts were on her side. If I were the OP, I would have felt aggrieved - by a capricious "girl" who doesn't take her own commitments seriously.
Perhaps some of you are right to call the OP out for referring to the woman fence owner as "the girl", but this woman has demonstrated her immaturity by agreeing and then going back on her word without a proper explanation. A mature person would certainly have done that! This woman didn't. If immaturity isn't the explanation, the only other explanation is that the fence owner is a liar, deliberately provoking upset!
What do these vicious posters have to say about that? Nothing - they conveniently ignore that fact, because they're enjoying the act of collective bullying.
The story is now clear for anyone who followed it:- the whole argument is based on a misunderstanding and ignorance of legal facts - now we know that the offending fence belongs to the woman across the back fence, and it's entirely up to her whether she wants to replace it or not, as long as it is not infringing on OP's property.
The OP will be doing the woman a favour by going round to tell her that the fence in fact belongs to her and she can do with it whatever she wants.
If the OP wants to build her own fence, of course she can - on her own property - all she has to do is to build it one inch on her side of the fence. One might say that the woman on the other side will benefit - all she has to do is remove her old rickety fence in order to get a new one. But that isn't quite so simple - she will never be allowed to paint it - it's not her fence and is one inch beyond her property boundary. Also, the OP could legitimately complain about any encroachment on her land - all the one inch of it - be debris, e.g. leaves, or pets, or even plants.
Therefore, in my view, it isn't really in the interests of either party to have a second fence which is entirely on OPs land. Just imagine "F OFF" could be painted on the other side of OPs fence for "the girl" to discover if she takes her fence down... only joking :-)
The OP would be wise to go over and talk with the woman owner of the fence, say that she looked up the title deeds, that the fence is in fact her property. She can then say that it's her choice whether to replace the fence or not - but if she doesn't, then the OP will build her own fence on her own land, and the "girl" or woman, if you prefer, will not be allowed to touch it, paint it, encriach on it in any way, and the only way she can reasonably do so is to maintain a second fence of her own on the boundary line. Which would be a waste... not green, not good for hedgehogs or other wildlife, etc.
OP could also say - if they can agree to replace the old fence with a new one - they could share costs and save themselves a lot of future hassle. If the owner of the fence agrees, get it in writing, call it a Party Wall Agreement (look it up). The OP could then produce a quote - but the fence owner might want to get more quotes, which is OK, as long as they can both agree on a budget and the look of the fence.
Ultimately, the OP is in a better position - if the fence owner strings her along, there's nothing to stop her from building her own fence on her side of the boundary.
Good luck OP.