What do I need to stay at my brother's wife's house after quarantine in Thailand?

The TM.6 isn't so important - but I suspect that's not what your're talking about. I think you're talking about TM.47

If you plan to be in Thailand longer than your original paperwwork/visa - you will need to 'check in' with immigration. If you are staying at a hotel they'll do this for you. If you are staying at private residence (or airBnB) you will need to make sure this gets done. That means you will likely need to do it yourself. Brother's wife can do it for you and if she is nice, she will do it for you. She probably speaks Thai and you probably dont.

My landlord made a photo copy of his passport and house-book (brother's wife's national ID card, and her house-book). My landlord scribbled on the page - OK for [your name goes here] to do complete TM.47, then signed. So.. scribble [TM.47 - clear to read] scribble scribble scribble all on a photo copy of a passport.

I showed up at immigration with the above and photo copies of my TM.6, passport detail page, and visa page. I was out in 15 minutes. Or they'll ask for some more copies of stuff.

This page explains better. The key is you only need to do this if you will be visiting immigration for any other reason in the future. If you plan to leave Thailand in less than 30 days or before your visa expires - this step isn't required

To do it legally you need a letter from her inviting you to stay, and a copy of her ID card.

Better to get a letter inviting you to stay and listing address and relationship details.

What visa?