Things to be aware of when making this repair:
1. The glue joint between the hammer head and hammer shank can be loose but when you wiggle the hammer head you don't feel it wobble. This will require a specialized tool to remove the head. You may also want to check for a loose hammer butt dowel, it has a similar sound, but the pitch of the click is a bit higher and more focused or precise.
2. When pulling the hammer off the shank, be careful not to break the shank (dowel).
3. After pushing the hammer filled with glue some will seep out. Do not let any glue come into contact with the felt that cushions the hammer shank (in the video this is green yours may be a different color) as it could cause the shank to be glued to this felt and potentially harden the felt causing an additional noise.
4. Though you can't see in the video, there should be a collar or glue around the hammer shank where the hammer is glued. Look at the neighboring parts to see what they look like. The better the collar, the less likely this will be an issue.
Be patient and take your time. This is one of the easiest repairs to make. Good luck!