Being a transwoman and going from he/him to she/her, I get that a lot. The longer people knew me as a man, the more difficulty they have. Some catch on quite quickly and just make the occasional mistake, some have to date not caught on yet, even though they are as supportive as they can be at the moment, given that they are going through something akin to bereavement. The longer they know you and the closer they are to you, the harder it gets and the more patient you have to be. Months, years, even.
Changing pronouns and first name is a big deal. It usually comes after a lot of deep soul searching in trying to figure out what you are and letting go of some of the cultural norms that have left their mark on you. You have made that decision and of course you would like everyone, especially close ones, to conform. And when they don't, it feels as if they are invalidating your choice and thereby an essential part of you. Frankly, it hurts.
But it does not work that way, close ones have to let go of those same cultural norms with respect to you, to view you as you are and not what you used to be and finally, to break with the force of habit. This is even more difficult when you are not at either binary end of the spectrum.
Now it is much preferred that "Jane" does this. You as a partner can chime in or provide tips, provided that you have squared this with "Jane". So where necessary substitute "Jane" with "I" and me.
Announce what the pronoun and first name should be
Do this gently and with some empathy for how difficult this is.
Alice? I understand that this may be difficult for you, but "Jane" would really like you to start using they, their and them when you are referring to them. It is really important to them.
Conversation may now go to 'Yeah but why?' or to apologies in advance.
Answering the why
Without saying that it hurts (leave that until there is no significant progress):
"Jane" is very sensitive about it. It has been a tough and hard decision for them. Yet the decision has been made and by using him/her instead of them it is like we are saying that they have made the wrong decision, even though we don't mean to. We must respect their decision and follow, even though it may be hard and will take some time to adapt to this new situation.
Redirecting apologies in advance
It's okay to mess up, it takes time to learn. Here is a tip, when you say him/her instead of them, just correct yourself. Don't apologize or only with the briefest 'him/her .. sorry .. them'. It shows you are making an effort and that is what matters.
This is part personal experience, part stories from others. Today my ex wife called me by my dead name first, then immediately called me by my assumed name. This is much less painful then having to listen to an apology, including rationalizations on why she messed up.
Announce intent to correct
Corrections on the use of the correct pronouns and correct name should be done gently, but brief. Tone is important here.
When you are corrected and that will happen from time to time, just acknowledge it and move on. Like in 'He/she did that then', 'They', 'Oh right, they did that then'
It is also important to reward good behavior. Smile when Alice uses the correct pronoun.
Eventually there will also be questions from Alice, aiming to understand why "Jane" has changed or chosen this path. These are natural and part of the bereavement process/paradigm shift. Simply answer them truthfully if you can and also make it clear that "Jane" is not unique in this and that "Jane" is not solely identified by their gender. But beware of Too Much Information as this may make it more difficult for Alice to come to terms with the new situation.
Above all, be very patient.
I wish I had all the answers on this one and give a foolproof recipe, but I don't as I am in the middle of a similar process.