To sew Mr. Fumble back together you need a sewing needle.
There's no needles laying around. A makeshift needle has to suffice.
This is the low-spoiler guide for "A sewing needle" puzzle in Lucy Dreaming. Unlike in normal walkthrough, you get gradual hints on how to solve the puzzle. Click the big button below to see your first hint. If the hint contains a spoiler, you will be warned about it before clicking.