If you'd like to request a replacement pen, go to Device service and repair. Sign in with your Microsoft account and select a surface or register a new device.
Once you select a registered device, you'll see replacement offers based on your warranty status and eligibility for exchange. To learn more about the Surface Warranty you can visit Surface Warranty Information.
In a classic MS move, the charger firmware needs to be updated but that requires another Surface. In the video below he talks about it at the 17:50 mark.