how do i transfer my data from my broken iphone to my chromebook

If the iPhone won't turn on, then there's no way to transfer any data from it. If you have any files synced in iCloud, you can log into your iCloud account from your Chromebook or another web-capable device and get to the files that way.

Regards.

If your iPhone won’t turn on, it’s data cannot be transferred to any device by any means. Repair the iPhone or give up. There are no options other than repairing it.