So this is where Adam says she will head to the nearby human kingdom for help. So it's a bit of adventure to get to the human kingdom and a bit more to get to the captail and a bit more to get to the royal palace. So the wrinkles start here. All sorts of things are happening all over the place, but Adam is locked into "getting to the human king". So the game rolls on and the princess gets to the human kings court.
When I say "She must wait until the next day" Adam gets antsy, maybe upset. Anything I try and do or make happen in the game, Adam is just like "the princess sits in her room. Can she see the human king yet?"
So the next day comes and the elf princess gets to meet the human king. Now Adam is all like to the king " I need help, when can we leave to get my kingdom back?" I try and slow it down with some role playing, but Adam just has the princess saying things like "lets go!". So I have the king slow things down saying he has somethings to do and will see the elf princess tomorrow with an answer. Adam is all antsy. It's late, so we end the game session here.
Obviously, we don't know exactly what is in Adam's head, but we might make guesses... Their play is consistent with someone who doesn't get what's expected of them, or who doesn't know what hooks to pull on.
You speak here about "slowing down" the action. The player pretty obviously doesn't want to slow down. They are laser-focused on their goal, and that's actually pretty reasonable for the character. What we don't know is
what else there is for the player to work with.
Some players are great at making up things from whole cloth to work with. Some, when given no path to follow, will say things like, "If I cannot see the king, I'm going to go down to the kitchens, find out what his favorite meal is, find out if there's any rare ingredients they'd need to make that meal, and go acquire them as a gift for the king..."
Other player's won't. They will wait for you to lay out pieces that are pretty clearly for them to work with, and start to work with them.
Rather than, in her meeting with the king, "slowing down with roleplay," (which isn't specific, but doesn't sound like the roleplay is directly relevant to the goal) you might have had more success with the king laying out the current political situation he has to contend with that the elf might engage with to win support for her cause. Or have the king start asking why he should help at all, or what is in this for him, so that the elf might engage with something that moves toward the goal.
If this were a dungeon crawl, you'd have laid out the physical cues of where to go next - the other exits from the room, or interesting features there to interact with. What are the metaphorical interesting features of the king? What other exits from the stall have been presented?
Or, if you expect, when faced with a blank room with no exits, that the players wield pickaxes and dig a new way out, you kind of need to tell them that.