Pamkir, I've never forgotten a place where I once worked (until I had to be transferred). It was a small office, with a warehouse in the back, where machines printed and inserted materials and where materials were stored on tall racks. I thought of it as a pretty dismal place and in a dismal location, to boot. But I loved it, mostly. And there was a corner in the warehouse (of all places) where employees were allowed to smoke. I worked in one of the crammed office parts, but when I took breaks and it was raining or snowing, I went back there and smoked. And they had a sign that said something like THIS IS YOUR SECOND HOME, KEEP IT TIDY.
Reading that sign and digesting it (like I did) was probably the first step for me when it comes to understanding how life and work come together, or need to stay apart. The sign smacks of "talking to children" instead of grown-ups and (my goodness) people that had gone through a university education. But I also think the sign talks to people, and people are ALWAYS children at least to a tiny degree if not a huge one, and they always carry their child selves everywhere they go, including their second home, work.
This Jane Do you are describing has been with your organization for at least 2 years--definitely her second home. And definitely a place where she can bring home into work, and all the child things (not "childish") with her. The workplace she shares with you and the team is not a place to have an open-book case study on her life experiences; it's not a nursing home, it's a business. It's her second home, and one to keep TIDY.
I definitely echo Karen's remark about an EAP, and where there is one lacking, she needs a person skilled in human interactions to explain to her that her personal challenges are not subjects that should stir any energies at all at work, as these are not good for the team at large. Personal dissatisfaction at the specifics around her teammates communicating with her during her absence are not open for further analysis, as her teammates are being relied upon to provide work for the business and not create a work exercise based on her personal business, all while they have shared a concern in her best health.
I do think providing her with "something different" is the right approach, as this is mind-activating and lends inspiration, but I would have a hard time defending this instinct to someone in the most senior position that would question why something of an "upgraded" nature has been placed in the hands of someone who is dealing with personal difficulties. (In the very end, that voice is interested in the health of the organization, whether the individual is eclipsed by it or not.) Work is not done when the project at hand is someone's below-standard husband or their rotten family or their bad health or their bad luck with their automobile. These things cannot be the "project" for your team, and they cannot abide by the TIDY workplace sign. And frankly, EAPs cannot abide by it either, because they do not operate on site--so intervention by HR or people skilled and in that capacity need to inject themselves in the situation.
My heart goes out to the person, but there are critical cases where the PERSONAL needs to be removed from what can become a "project" and Jane's issues are not a team project. If they already are, that needs to stop.