I'm not promising you will stay sane, here are just a few things that have helped stop me climbing the walls -
1. Start off with Good intentions
What day is it? Monday? Who knows - stop the days from melting into each other by assigning Monday as your "back to work/school etc" day. Get dressed, wear your best work outfit. As the week goes on you may digress further and further into pyjama territory, but that's ok, at least you can start again on Monday.
2. Keep a guilt diary
Write down when you feel guilty about not being productive. Then time-travel to 5 years in the future, look back on yourself when you were in lockdown in the middle of a global pandemic and assess those feelings for what they are... ridiculous (but human). Perhaps you could write a note of encouragement from your future self back to your past (i.e. present) self, that says, "hey dingus - give yourself a break" or words to that effect (my future self is a bit of a straight-talker).
3. Take your gosh-darn walks
If you can, take your walks. This might get worse. Don't be the doofus who regrets not taking her last chance to walk outside because she wasn't "feeling up to it". Stay safe and take all the precautions. But if you can, go now. Walk.
4. Permission to switch off - granted
If you need permission to stop watching the news, talking about the news or going on social media at all, here it is. Go for it. Half of it might be fake anyway - like those Dolphins in Venice? Apparently that was never a thing. So sad. Let's take a break.
5. You can eat what you want... if...
I just finished reading How Not to Diet recently, so forgive me if this is on my mind during lockdown. One point I'm using at the moment is "Breakfast like a King, Lunch like a Prince and Dinner like a Pauper" which is no longer just an old myth but scientifically proven. And now we have oodles of time in the day to eat our biggest meal at breakfast or lunch and it's more realistic to stop eating after 7pm so I'm taking advantage of this new rhythm (and not feeling guilty about my obligatory 11am cookie).
I've read so many 'stay sane' articles that all say the same thing: keep a routine, take up a new hobby, meditate, use this time to achieve all your goals, read a new book a day, all these ridiculous things and I didn't really want to add to it. But the above isn't general advice, it's stuff that has helped me, is still helping me currently as we enter week 6 (?) of lockdown and I'm semi-smartly dressed (it was Monday yesterday) and about to go on my "post-work" walk. If it helps you too, that's nice. Best wishes to you and your family.