There are many options for laundry in tiny space. From laundry mats to all-in-one units, to your mother's house. This is the setup I've been using since before moving into my tiny house actually. I used it while living in a motel as well. It is more manual than many options, but I don't mind this.
I have a wonder wash as my washing machine. You just fill it with clothes, water, and your choice of detergent. I use soap nuts since they do not irritate my skin and keep my water totally safe for dumping in my garden. More about them in my post here.
Then you simply close the lid tightly and spin the machine back and forth to tumble everything. Drain and add fresh water and repeat to have a rinse cycle. When it's warm, it seems most handy to do all this on my little porch. In the winter I stay inside.
To drain the machine, you stick this little spout in, and it drains from the bottom. I collect the water in a bucket and pour it right on my plants.
Once everything is rinsed, it goes in my Nina Soft Spin Drier. This works like a giant salad spinner and gets an amazing amount of water out. Clothing is nearly dry to the touch when it comes out, but still a tiny bit damp.
Out of the spin drier, bigger items go outside on my wash line like the very first photo, and little things or delicates hang on the folding rack I have inside. At least with the dry air we normally have here, things only need to hang for an hour or so to be totally dry.
And that's my laundry day here at Fy Nyth!