You can usually remove the hard drive or SSD, whichever that unit has, as long as it DOES have a storage drive and not some kind of flash storage (Would need to know model numbers in order to know that). Usually most laptops, not counting some Chromebooks or Netbooks, have either an SSD or mechanical hard drive which can be removed.
Then you can connect it to the motherboard on a desktop, or even to the laptop you have now that does work IF it has room for another drive. If not, then you would need to get some kind of external enclosure for it to connect via USB to your laptop and move the data off.
Something like this would probably work...