I'm wondering if there is a connection to her anemia and if she is seeking out 'iron'. Maybe from minerals in the water itself or from the faucet metal? It could be the combination of the water coming right from the tap that smells more like iron than when it's sitting in her bowl or in the gerbil water dispenser. I've heard of cats licking certain things when they are depleted of something in their bodies and maybe this is what it's about. This may not help one bit but you could try and find an iron cooking pot or pan and fill it with the tap water and place it near her.

If that doesn't work, could you create a ramp up to the sink? It might not be as challenging as a step ladder for her as long as it's not too steep.

Praying that Puddy begins to get better.