Hi there! Thanks for your message, and I'm so sorry to hear that you were having issues. The bust has been giving some players trouble, and it's very high up on our list to address with our next update. If you weren't able to move any of the pieces it often works if you exit by pressing E and then re-renter the cleaning minigame by interacting with the bust again. You aren't meant to rotate the bust when it's broken, so all of the pieces can be put back together from the front. It can take a little bit of rearranging to make it click into place.
I did notice that yours has a very similar premise, and I can't wait to give it a shot!