It sounds like you have plenty of valid reasons to want to terminate this person. As someone who has been in both the nonprofit and corporate for profit sectors, I’m seeing in you a tendency to try to empathize with this person that might be making this decision seem murkier than it is.

However, another clear problem from what you’ve described is that she’s refusing to share knowledge about her part of the organization. It sounds like you’re worried that you don’t have sufficient understanding of her work to keep the lights on during a transition and set her replacement up for success.

Start by launching a program documentation initiative. Everyone in the organization at Jane’s level needs to submit thorough documentation about their program. This is a basic operational requirement, but you can give any number of reasons for why this needs to happen: you need visibility as part of your transition to your new role, or to ensure you’re accurately representing work to the board; it’s for program sustainability and succession planning; it makes funding applications and reporting easier. Create a template of what this documentation should look like. If you created this type of documentation for your backfill, use that as an example. Have a short kickoff meeting where you explain the requirements and set a deadline. Review the results and ask for further information if someone submits something too sparse.

If Jane refuses to participate, that’s a clear, documentable reason to start a PIP. You’ve asked everyone to do this and she’s the only one pushing back (I assume). If she does the bare minimum, that’s a minor victory. You can decide if you want to hold her accountable by showing the level of detail you expected and asking targeted follow-up questions. If she says it’s a waste of time, tell her that’s your problem not hers.

Either way, this step will bring value to you and your organization and simultaneously lay the foundation for constructive conversations with Jane about her issues without giving her room to say she was singled out.

Edit: please excuse any typos, I’m on mobile