Sorry to hear your mum's still struggling with this, especially in light of recent events.

I gave my family 6 months before I started putting my foot down with them. That's plenty of time for them to get used to it, and anything longer is just sheer belligerence on their part. Look at it this way: we as a society already have an established procedure for people changing their names, in that women often change their surnames when they get married. When that happens, she announces her new surname and everyone else is expected to just get used to it. On average, it takes people about 6 months or so to get used to the fact that Miss Smith is now known as Mrs Jones, so we should expect the exact same respect and courtesy when we change our names.
So that's the benchmark I used. I corrected my family every single time for the first 6 months, and after that I warned them that I do not and will not answer to my previous name & pronouns. I told them that the conversation would be over if they slipped... and I stuck to it. So the next time it happened, I just calmly ignored them & walked out of the room. If they tried it on the phone, I instantly hung up on them. Eventually I told them that if they ever call me 'she' to my face, I will lift up my shirt and show them my top surgery scars & hairy chest, so they can see for themselves just how much of a 'she' I am. They got the message pretty darn quickly.

Now they only misgender me behind my back (which is really douchey of them tbh) but at least if they're not doing it to my face, it doesn't have to hurt me. I count that as a win.
I honestly believe that the longer we put up with people 'slipping up', or the more we try to make accommodations for people who claim to be trying but aren't really - the worse we make things for everyone involved, including them. If we allow them to persist in misgendering us, we're telling them that their old view of us is still valid and that we're in the wrong about who we are. You may think you're doing your mum a kindness by giving her lots of time to adjust, but 6 years is waaaay more than enough time for her to get her head around
anything (she could earn a PhD in that time!) so it's not time that she needs to break her out of this cycle. What she needs is a line drawn in the sand, beyond which she may not go. Do it kindly, but tell her in no uncertain terms that her fictitious view of you can no longer be humoured and she has to use the correct name & pronouns every time she speaks to you or about you.
You are
not a shameful secret, and you need to tell her to stop treating you like one. What's more important: what the neighbours think, or her daughter's wellbeing? She needs to accept that she has a daughter named Jayne and that's that - so when she talks about you she needs to talk about you in those terms at all times. As for that dinner thing with the menus? I know you were upset & it probably wasn't the right time, but under different circumstances I'd have stood up and got them myself, making a point of saying 'well as I'm a woman too I'll fetch them!'. (Although I do wonder why it should be a woman's job to fetch a menu - what century is your stepdad from?).