This is my take on the day or two after the film ended.
Doctor Finkelstein had been in his study early in the morning when Jewel informed him of the Pumpkin King's sudden arrival.
"Ah, Jack Skellington," he greeted plainly, while Jewel wheeled him towards the table then. "You must forgive me. I didn't expect to have company at this hour."
The elegant-suited skeleton nodded in return, settling himself in the chair opposite of him. "Well, thank you for seeing me anyways, Doctor."
Jewel turned for the stove placed across the room and she returned with a small iron hand kettle. "Would you care for a cup of black tea, Your Highness?"
"I guess one cup wouldn't hurt." Even if Jack hadn't found the offer necessary, he still wasn't about to be rude out on a business run.
Once the china was set out and the beverage was poured, Doctor Finkelstein had dismissed his ward in order to gain a bit of privacy.
"So...tell me Jack, what has really brought you here on such short notice?"
"Well, Doctor, I have come on Sally's behalf."
The doctor paused in mid-sip, refusing to display any sort of bold reaction.
Jack's skeletal fingertip began to leisurely trace the rim of his cup. "...Judging by your lack of curiosity, I figure you must be disappointed in her?"
And yet, when Dr. Finkelstein processed this statement, his next words weren't quite out of promise or disagreement. "I must admit, that she did not live up to my initial expectations. Nevertheless...I don't create anything unless I'm not confident about the potential it could have."
"And that is why I am here, Doctor—I plan to take full responsibly of her." Taking a light swallow of his own tea, the Pumpkin King allowed it to flow down his ribcage before he stood again to leave. "You do not have to concern yourself with her welfare any longer."
The King was feeling slightly guilty at this moment discussing Sally so impersonally like this. But, he also knew Doctor Finkelstein was the most methodical-minded citizen in all of Halloween Town. Meaning that if he wanted things to work out in his favor, he needed to play his cards correctly. And the doctor was never one for supporting the heart's desire over practical statistics.
"Creations are intented to have a true purpose, Jack." the doctor added strictly, as he whirled his head along with Jack's fluid motion. "She has not found hers yet. Her staying with you may not be the best thing for her in the end."
However within moments before exiting the study, Jack stopped merely to reassure him. "In truth, I think she has."
Doctor Finkelstein released a sound resembling a gentle scoff, "Which is?"
"...My true happiness."