"And yet you wasted a whole morning on being useless and mean."

"Mm. Yes. That was a mistake, an error of judgement, something to which not even I am immune."

She pretty clearly doesn't trust that. 

She is not required to.

Thence, to magic. The basis of magic is the power that flows forth from what are called wellsprings- fonts of mystic energy. Everyone has a wellspring within them, and attuning to that is the first step. But there are also naturally occurring wellsprings out in the world, spewing forth undirected magic that saturates the environment and eventually, coalesces into ley lines. When one's own wellspring is insufficient, these natural wellsprings can be called upon to augment the amount of magic available.

Like some ancestor did with her family's castle?

Yes. That was an example of what can be achieved with ritual magic, as such augmentation is called. A wide-scale, long-term, powerful effect.

Lily used it. Does that mean she's already attuned to her wellspring?

Partially, at least, if only on a subconscious level.

How does she attune all the way? 

Mindful self-awareness. Meditation, traditionally.

Lily's never meditated before. 

Then she'll have to teach her.

What's the first step? 

Get comfortable, breathe evenly, quiet your thoughts...

She's having the most trouble quieting her thoughts. It's unclear how much of that is a naturally busy mind and how much of that is her still being on high alert from the Demon Queen's presence and how much of that is her mind jumping back to recent distressing events (which is basically all that happens when she tries to clear her mind; the faces of her dead family, the events of this morning, the threats hanging over her all rush to fill the void left by other thoughts, and she can't get those to go away, and the more she tries the louder they get).

Some people find it helpful to occupy their body with activity, rather than sitting still. Stretches, kata, that sort of thing.

She's still not willing to get anywhere near the Demon Queen when weapons or combat are on the teaching table. And stretches sound boring. 

Then go for a walk or a run.

...Might need a run. She also used to ride her father's horses...

(Especially if her parents weren't there to scold her for giving their most spirited horse enough loose rein for a gallop.)

Perhaps they can investigate the stables tomorrow.

Where can she run now? 

She'll show Lily the track.

She runs like something's chasing her, with all the boundless energy of an athletic kid. The beating of her heart, the desperate gasp for air in her lungs successfully drives the thoughts from her head, and her entire world becomes the track before her, the movement of her muscles, the energy within her...

And there, beneath the pulse of blood and heart, a slow, faint, seeping trickle with the whisper of intimation that it could be more...

She pulls air into her lungs, and her heartbeat rings in her ears, and she opens herself to that trickle and she runs and runs and runs faster -