Go to account.microsoft.com/devices, sign in with your Microsoft account, and choose the device that needs service, select Next, and then follow the instructions to complete your service request. If it isn't registered, select Register device and enter the serial number.
The portal will show whether the device is covered. If it's outside the Limited Warranty, your product may be repaired or replaced depending on the available out-of-warranty offers, which vary by product and country. For a Surface Pro 8 that often means a paid service fee, or a refurbished replacement unit. The cost page (support.microsoft.com/warranty) lists US prices for your model.
How much does out-of-warranty service cost for your device or accessory?
The other routes are Microsoft in-region repair and Authorized Service Providers. In some regions Microsoft also has service centers that provide in-person support. For a local option, check the Authorized Service Provider list on Microsoft's support site.
How to get service or repair for Surface
If the Surface turns on while plugged in, backing up and erasing your data before shipping is recommended.
If the portal doesn't offer a battery option for your unit, contact Microsoft Support through the Contact Us page or the Get Help app and ask them to open the order.