From personal experience...
First of all, allow me to say how sorry I am this happened to you. I know exactly how difficult caring for your partner while also having to do all the childcare as well in this way is.
For context, my kids were a little older than yours when I had this experience, but nevertheless I was out of the career world for over four years.
Does this even need to go on LinkedIn? I have never addressed my career gap on LinkedIn or even on my CV when applying for roles. I was asked about it in a couple of interviews but I never felt that it really held me back. When I was re-entering the job market, I knew I'd not kept my skills to the level I would have if I had been working, so I was ready to take a pay cut to get back in and prove myself.
When I accepted a role (only the second or third I'd applied for once I was looking again) only my line manager and their boss knew why I'd taken a career break. I didn't discuss it with anyone else in the company for over a year. I guess I was somewhat concerned about perceptions and I didn't want to be "that guy" who talked about nothing else. And I certainly didn't want it to be all I was asked about for ages after I joined.
You don't say what the outcome has been for you and you family, nor should you have to. Unfortunately our outcome was not good, so maybe that coloured my view about speaking about it during the early days back.
I hope it has been a better outcome for you all!