Hi Josie! My name is David. I am a user of Microsoft products like you, and it is a pleasure to greet you.
To better understand your issue, when you mention that your screen is completely broken, does this mean that the screen doesn't display any image?
If your screen is not displaying any image, it's possible that your TV is not automatically detected as a second screen by Windows. This may be because, even though your screen is broken, Windows is still detecting the device physically. Please try the following steps. After turning on your PC and waiting for a few minutes to ensure it has fully started up, press the "Windows" + "P" keys. This will enable the screen selection. Press the keys twice and release them. Repeat this process until the extended screen or second screen allows the image to be displayed on your TV.
Please ensure select the HDMI input source you've connected your laptop to. This can usually be done using your TV remote.
Let me know if the shared information has been helpful. Regards.