She finds the instructions in the back of an old book that must have been tossed across five or six dimensions over the course of its lifetime. It's in Latin, rather than a demon language. She knows Latin. She's pretty sure Imrainai doesn't know Latin, which is presumably why she didn't confiscate the book.
The book says you can follow the instructions to summon an angel, a fairy, or a demon. (She's really unclear on where the first two categories have been all this time, but whatever.) Angels transform stuff into other stuff. Fairies move stuff. Both of these sound potentially strategically useful, given enough time to strategize. Demons, though? Whatever kind of demon this is talking about is an entity that can make stuff. A summoned demon could maybe make her a cell phone. There's maybe something to be said for sticking with the devil she knows, but the devil she knows is actually pretty terrible? So hey, lots of room for improvement even within the evil fiends category.
The instructions are clear that if you're going to summon anything, especially a demon, then you gotta make sure that your circle has bindings. The book gives some examples, and suggests other precautions one may want to take. It strongly, strongly suggests gagging demons, because demons are evil and stuff, and talking to evil things isn't a great idea, apparently. On the other hand, she has practice on that front, and she doesn't have a lot of stuff to trade away or a lot of time to do it in, so playing twenty questions with a demon sounds possibly more dangerous than just talking to it, under the circumstances.
She spins up a binding. It's on the conservative side of the book's suggestions, but the gag only keeps the demon from speaking out loud, not from writing or signing. When she's moderately sure that she knows how to make a summoning circle work, she goes to bed in her room as per usual, turns out all of the lights, very quietly and stealthily tapes some black construction paper over the hidden cameras, piles a blanket in front of the crack under the door, turns on a flashlight wrapped in tissues that gives her just enough light to see what she's writing, draws the whole circle out on her floor with chalk, and thinks about how if this works then she might be able to win.
In the quiet and not-quite-blackness of her room, Suzanna Teller summons a demon.