Unable to update my Apple Watch Series 6

If you did manage to update the watch you would need a new phone.

OS 9 needs IOS 16

OS10 needs iOS 17

OS11 needs iOS 18

iPhone X cannot go beyond iOS 16

You cannot have it updated through your daughters phone, because the watch needs to be paired to her iPhone in order to do that, and it is not her watch

You can try to update from the watch itself

Update directly on your Apple Watch

If your Apple Watch has watchOS 6 or later, you can install updates without your iPhone:

1. Make sure your watch is connected to Wi-Fi.
2. On your watch, open the Settings app.
3. Tap General > Software Update.
4. Tap Install if a software update is available, then follow the onscreen instructions.

If this is not working, you might want to restart the watch to see if the issues persist