Having a broken screen on a surface pro means you can remove the broken screen and not worry about breaking it. SO the old screen is glued on to the base, once the glue is hot enough the old screen can be removed. or if you like just pick it off as you have a new replacement surface so you say.
What we are getting at is to remove the SSD drive of the old unknit and purchase a mini SSD to USB device, then plug it in the new surface pro or another computer and remove your data. If you do some research you will find the adapter mostly empty for building a fast thumb drive and with usb output for the plug on your surface.
I have used a hair dryer and a gloved hand then a guitar pick or similar device to push in and gain access to lift the old screen. They have special heat guns and suction cup to pull on the glass but a little work and the glove so you do not burn your hand. I also have incorporated tooth picks to keep the glass away from the glue as you pull the glass up so it does not contact glue back down.