Jack's other copy was in Euphosine's chamber. But since he was just a copy, he couldn't search around with his hands. He could only use his eyes. From what he could see, nothing in the chamber looked like a key.
The chamber even seemed like it hadn't been used for quite some time.
Jack guessed perhaps for someone who had gained the power of a Goddess, sleep was no longer necessary, and so was a bed-chamber.
There was a cupboard, a dresser with a small chest above it, and a box draped with a cloth. These were things that might hold valuable things, but he couldn't do anything to them as long as he was a copy.
His real self was at his last one hundred lockpicks. He kept on thinking the next lockpicking would succeed and ended up spending several hours lockpicking Talia's chain. Since they entered this nexus world, more than five hours had passed. He was not confident he could open the lock at this rate.