Congrats on finding a house! The only word of caution I would have about a basement is if you think that you would spend a lot of time with Loki if she is down there? It might be nice to put a couch and TV or something like that so that you can sit and spend time with her. Also, does it have any windows so she could get natural light and fresh air sometimes?
I actually use exercise pens for cages, I like them because you can change the shape and move them around as needed. With an exercise pen, you could even bring it upstairs sometimes and let her play up there. My bunnies really like to tear up cardboard so a cardboard play house would be fun, and if she isn’t so good at the litter training, you probably don’t want to build her play house with wood floors. I did that for my bunnies and they ended up peeing all over one of the floors and ruined it because its tough to get the pee out of wood
As far as the concrete floor, I think you will really want something softer for her to use, the problem with vinyl is that bunnies slip on it easily, mine have slipped and fallen on the bare linoleum in my cage floors, so I bought seagrass mats to put over it. You can also buy cheap rugs at discount stores like Big Lots or Marshalls. Have you looked at litter training tips to try to get her to use her box more? Have you tried this?: I Buy a few inexpensive towels, and then as soon as she pees on them, mop up some of it with a paper towel, and put the paper towel in her litterbox, then replace the towel and put the other one in the wash.