He isn't apparent that second day.

That would be too much to hope for.

The next step is then to seek a formal audience.

Not impossible to get, but a lot of people want audiences, many of them with a concrete reason - diplomats, those formally seeking judgement on a civil dispute (which there's a separate channel for, generally based on recommendations from lower courts), petitioners... There's at least some effort to triage urgency and need; if she doesn't state a reason, her case will end up in the general pool of petitioners.

She presents herself as Ellisaria, a representative of the dragons.

That gets her placed pretty high in the pool of diplomats; the Steward will be able to see her tomorrow, or his son later today.

She will see the son.

Her appointment is in a few hours, after the midday break.

She waits patiently.

And, eventually, she's allowed into his presence - guards present unobtrusively, and she's not to approach past the table or the black line down the room's middle unless invited.

He's standing when she enters, behind a large table and next to a rather notably unadorned wooden chair. There's a chair clearly for her across from him. It has the feeling more of a cozy office, like where a merchant or general or judge might meet for business, than anywhere a king would host petitioners. (The white tile floor does, indeed, have a line of black scroll work running across the middle of the room, underneath the side of his desk nearest her.)

"Ellisaria, was it?" he asks, smiling slightly.

"That is correct. And you would be Ecthelion."

Nod. "Yes, Ecthelion the Second, son of Steward Turgon." He gestures to the chair across from him, and takes his own chair. "Feel free to sit, if you'd prefer."

"Thank you." She sits down.

"Now, you're a representative of the dragons, is that correct? Which ones?"

"All of them. Or any you are likely to hear of, at least."

He regards her for a few moments, expression still friendly, but more serious now, too. "What would the dragons have of us, then?"

"We share a common enemy. I have reliable information that the lord of Mordor has returned, and is once more beginning to gather his strength."

His expression goes grave. "Something we'd feared."

"Then you are not willfully blind. Good."

"It's hard to be, here. Though it does hearten me to hear the dragons consider Mordor their enemy; one piece of good news, at least."

"Mordor should be the enemy of all who value this world."

"Sadly not a universal position."

"So I have found. Which is why I have chosen to approach you."

"I am glad you have. Do the dragons wish to discuss alliances beyond such warnings?"

"Yes. My primary desire is a strategic and military alliance, to contain and eliminate the threat posed by Mordor."

"Understand that I can't make guarantees of an alliance today - such negotiations are always long - but we're not fundamentally opposed to such a thing."