That's a tough one, if these are public torrents you may be stuck having to search for the torrents one-by-one and hope you can find their associated .torrent files to re-seed the torrents.
You do need a .torrent file, or bare minimum a magnet link to an active/non-dead swarm, to be able to re-seed torrents with existing data. Magnet links without any current peers on them are a bit useless for this task unfortunately.
If these torrents are from private trackers you can check the private tracker website(s), many of them have a feature to allow you to re-download all the .torrent files you previously downloaded from the tracker.
Easier with the model # of the Thinkpad so people can look up the specs, but working off what you posted and the pics..
Have you tried plugging in the 2.5" SSD there & then boot into the BIOS to see what it detects? (your 3rd pic, the slot on the right looks like it could take a 2.5" SATA drive) Carefully... just as a test run - Like the other comment said there isn't a drive cage or anything there so you couldn't really run it with a 2.5" drive full-time until you can rig up something to mount it properly.
This particular Thinkpad may have only been set up for M2 drives so getting a 2.5" drive working there could be tricky, I'd just get a compatible M2 drive to install in there since you know for sure that's what the Thinkpad was configured to work with previously.