I recently rewatched Down the Rabbit Hole and Bad Blood andI noticed something strange. When Lydgate is preparing for Alice's lobotomy in the first episode, one of the things seen on the nearby table is a syringe. Now, in Bad Blood, Jafar uses what appears to be the exact same syringe on Edwin to extract some blood. So, maybe Jafar took the syringe from Bethlem for some reason when he was there (maybe as a souvenir because he's never been to Victorian England before). This would be rather ironic because Jafar used the syringe on Edwin and Edwin is the one who originally suggested Alice going to the asylum.
At first I considered the possibility of a prop being reused (kinda like how Bo Peep's two bodyguards wore the same red capes the Tweedles occasionally wore), but because both Jafar and Lydgate had access to the item, I don't think it was just reused.