Check that your USB cable is plugged directly into the Finch 2 charging port as shown below. If the USB cable is plugged into the micro:bit the Finch 2 will not charge.
If you have plugged in Finch correctly and it's still not charging, there are two possible reasons it is not charging. One is that the battery is not seated properly in the holder, so it is not turning on. To correct this, please open the battery door and push the battery into the holder with enough force to depress the springs in the Finch. You can see this procedure in this video.
The second possibility is that the battery is no longer holding a charge. We can test this by switching batteries with a working Finch using the same video above. If your battery is no longer holding a charge, please email firstname.lastname@example.org to get information on how to get a replacement battery.