The question does not, as some answerers have assumed, say that 'Jane' is uncomfortable talking to people. It says she is an introvert. By the definitions of the category, precisely half the world's population at any given time are introverts, and not only those awkward programmers who fear to leave their closet to even make the short trip to the bathroom. The OP describes a competent introvert, not a scared child who grows nauseous at the idea of looking someone in the eye.
As @Lilienthal answered, much of this question is really more about how experts should be treated in an organization, and less about introvert/extrovert. But there is obviously some some value in taking someone's personality and preferences into account when managing them.
My suggestion for introverted experts involves a modified 'gatekeeper'.
Strong extroverts feel trapped behind a gatekeeper, but even so will often use them just because their time is valuable. A traditional gatekeeper is a receptionist screening phone calls for the executive, so they can instead have time to walk the cubicals and hang out at the water cooler (hopefully as tools to do their job, not slacking off). Your 'Jane' doesn't need that. What she needs is a single (very rarely multiple) contact point as a go between. This person will be the contact point for other departments. I would suggest identifying a junior level programmer (the question suggests that 'Jane' is a programmer, insert junior level [whatever] here) who is excited about the idea of making a difference and a name for themselves (usually young, could be introvert or extrovert) who, at the beginning, does your dirty work (commenting code, whatever, just learning what everyone else is doing, how well they do it, and what the workload is), and slowly (or quickly?) expanding that person into your 'marketing department'. Send that person to other departments, into their meetings, introduce them to your counterparts and clients outside of your team, and tell them that this person's function is to find out how you can do more work for them (not to commit to it!). You know some other managers will balk at an outsider entering their 'fiefdom', but that is your boss's job, not yours. What you are doing is both marketing Jane by going to the customers, and saving her valuable time.
This method allows Jane to have control over who approaches her (we all want, introverts want it more), and limits that number of people to a small group, while still doing what your boss asked.
How does this work in practice? Depending on your environment, this 'gatekeeper' should be told that they are (truthfully, of course) either being groomed for management, or being mentored by Jane. Regardless, this person should be sat down by you and Jane, and told in no uncertain terms that they are not Jane's boss, and quite the opposite. Just because a receptionist controls an executive's calendar does not give them the right to tell the executive which meeting is more important, even when she has to make judgement calls and basically do just that on her own. Jane must have control over who this person is, either by helping to identify them, or say in having them reassigned and a different person brought in, if they annoy her too much.
Regardless of mentorship or leadership track, the junior level person would have a standing meeting with Jane, eventually daily, but at first likely weekly. They would talk about what Jane is working on, what Jane needs from 'the outside world' (more detailed requirements documentation, additional resources, lunch?), what new projects are on the horizon from different departments and which ones Jane would enjoy, how long it might take Jane (or the team, she probably knows the team's capabilities at least as well as you), and hopefully Jane also gets excited about mentoring the junior person in their own programming (or whatever) projects. That junior person will report back to you their notes from these meetings with Jane right away as well as notes from any other departments they have met with,
One of the specific requested benefits here is that Jane will realize that, because of the volume of work that is out there, she can't spend the kind of time she wants on each project... or even better, may come up with ideas on her own of how she can. And all this will happen by people (you and 'junior') coming to her, not her going out.
Other benefits: your junior staffer gets their own stuff done much faster (especially if this is mentorship), as well as getting career improvement and advice. This will take from your time, and it will take from Jane's time (set parameters how long and often these meetings take, giving Jane the opportunity to extend if she enjoys the mentoring), but it does what your boss wants, and at a fraction of the price of what it would cost for Jane to go market herself, even if she weren't introverted. And perhaps most importantly, it shows Jane that you and your company trust her, value her, and see her as more than just the amazing production machine that she already knows she is.
Caveat: most important of all of this, though, is to actually talk to Jane. Ask what she wants. If she is cagey, dismissive, or passive in that conversation, then she is probably not the person that your boss thinks she is, and your boss's whole idea will backfire. Be the manager that your boss expects you to be, and give them this information up front. Also in this conversation with her, use your skills as a leader/motivator to get her to be excited about whatever idea (gatekeeper or whatever) you come up with, and create the method by which you and she will continue to give feedback to each other about how this is going.
Now, frankly, what is really going on here, is your boss just gave you the stepping stone to managing other managers. Probably intentionally. Own it, be grateful for it, and realize this is bigger than figuring out how to talk to an introvert. Managing experts with egos and increasing the business and notoriety of your shop/department is exactly what executives do all day. Welcome to the rest of your career. Good job starting it by asking for help.
