How to teach so that your student doesn't feel stupid?
The idea is that I somehow make it sound... too easy? Like it should
  be obvious. But when it doesn't immediately click, it makes the
  listener feel stupid. And that's when learning stops and inecurities
  kick in.


This reminds me of the first time I was teaching an intro to programming class and I was going through a live demo of how some code worked. I mistakenly had an error in my code and got a compilation error. One of my students gasped and said, "You get those too?!"

Previously, it hadn't occurred to me that students thought that people who are "good" at programming didn't get errors. They thought there was some inherent quality that I had that prevented me from making errors like they did.

I'd recommend modeling the problem-solving process. Try doing one of the problems yourself and just let her watch and ask questions. Think aloud and describe everything that goes through your head. Even try explaining different approaches you might consider in solving the problem. It helps demystify the problem solving process of programming when novices get to witness your thinking (and debugging) process.