.. Getting the clock to operate again

You have to find a good reason to do that, first. Search other areas until you find a reason to fix the clock.

So, you got into Santa's castle and talked to Idril about her breaks?

You should now talk to the clock guardians. Maybe Chestnut is more understanding about Idril's situation.

Unfortunately, one cog is missing. Also, half of the cogs are with Melsonict.

Then, solve the cogs puzzle.