B737-800 Captain reported an air turn back and over weight landing was caused by an engine anti ice valve failing to operate.

Narrative

DATE at XA:14 AM the event occurred on the departure on [Runway] XXL at ZZZ airport where the weather was overcast and around 7 degrees Celsius. At about 1;100 feet AGL; we encountered icing conditions with a TAT of around 6C where I selected the engine anti ice on. After about 30 seconds after selecting the engine anti on we noticed the 'eng anti-ice' light illuminated; which indicates a engine core valve has failed to open. We followed the QRH procedure where it told us to avoid icing conditions which we were encountering. At that point we were assigned higher which put us in a layer that keep us out of icing conditions for a period to make a decision to return to ZZZ or continue to destination. With weather over the mid Atlantic and possible icing conditions in view; we decided the best course of action was to get below the cloud layer and return to a long runway at ZZZ. We decided to land overweight giving the runway margin we figured in our 'landing app'. We called company; informed FA; the passengers and informed ATC that we were overweight to [have] emergency vehicles standing by at the end the runway. The landing was made by the First Officer which was smooth and right in touch down zone. We let it roll to the end with min braking to avoid high brake temps. Command One checked the brake temps and followed us to the gate.I would have advised ATC myself; I was under the impression that when ATC asked if we were overweight and need emergency vehicles standing by. Next time I will transmit the 'pan pan pan ' immediately.