I'm disappointed at how many people want to go a legalistic or aggressive path as their first play; no wonder lawyers are kept busy, with people being so poor at conflict resolution.
You seem to know that the owner is an elderly lady. I doubt that this is part of her grand retirement plan, to try and rip you off for a fence.
I think you should try seeing if you can meet with her in person, and either let her tell you the story, or just open with "I just wanted to talk about the fence..." and see what she says.
Whatever she says, you will have a much clearer idea of the appropriate response and you've lost nothing. Hopefully, if - as I expect - she just wasn't aware of the requirement to issue a formal Notice etc., then hopefully by taking this route you can come up with a reasonable compromise and keep the relationship intact. (Which you indicate you need, and which I think is always a good idea, 1) just in case you need something further down the track, and 2) just because it's nice to have good relationships.)
If you do go down the legal route, I suspect that eventually, a magistrate (or whoever hears it) would find, in equity, that you're obligated to pay half of a reasonable price, or ~$33/m (if the $66/m is a quote from a licenced tradesperson who issues a tax invoice etc.). But if you're obliged to go down the legal path, I doubt you'll retain the good relationship with the neighbour.
If she really had no idea she had done the wrong thing, and had no idea that the quote was high, then you have to consider whether the <$50/m differential ($81.50/m vs $33/m for your half) is worth damaging the relationship with your neighbour over.
