I think, firstly, you need to start working on yourself, ask your partner to help you navigate through this difficult time. Proactively find ways to get your productivity and passion to learn back....Get a standing desk if you have not already, personally, I found standing desk make me focus more. Ask others to help pick up kids or drop off so you can have solid 6,7 hours working and refocusing. I find it hard to juggle between kids and job as well.

On the "should I tell my boss" note, I would do that after trying everything else and ask whether it is possible to take temporary 1 week off without pay to get yourself back on track. Since you barely know your boss, I think you should not focus on how it affects your work yet and just explain how it affects your family everybody has family so I think your boss would resonate.

I like to look at it this way, you are hired because you have the skill and proved to be better than other candidates and the company took time and effort to find you, I trust that they will want to work with you through this instead of abandoning you.

Last but not least, I find that for every job, you feel a little loss at first but to truly integrate and feel like you are part of the team is to start having some wins (big or small) so you should look around, find taks that you can accomplish or want to learn/work on and actively request to do those. It will help boosting your confident and make working more fun imo.