Dual-HDD laptop crashed - trying to salvage the data

Hi folks, I'm at my wits end with this problem and am desperately seeking advice. So far, after scouring the internet, I haven't been able to find a solution that works for my case.

I had a 5+ year old MSI gaming laptop that burned out from what was probably a short circuit, based on my investigations. I assume it was either the MoBo or PSU that was fried but, since I was due for an upgrade anyway, it was an opportunity to let go of the old and buy something new. I did have a critical data backup, but let's just say it wasn't recent enough and I don't find it sufficient (lesson learned).

The old laptop had a dual HDD setup (2x 500GB SATA, model WD5000BPKT). It was a 2x partition with the primary partition running the OS (Windows 10) and the secondary just acting as storage.

I have a new laptop now (also running on Windows 10). I bought an external adapter for the old HDD's to connect via USB, but they are not properly recognized and require formatting to access, which I assume is due to the partitioning mess. Now I fear that the only way for me to salvage the data is to build a bare-bones box with the same dual HDD setup, which I'd like to avoid for cost reasons.

Any advice from the experts on what I can do here? Very much appreciate any help offered!
 
Why not just use system SATA on a borrowed PC to add the 2 old HDDs as slave drives? Copy to cloud or DVD or USB flash. Should only take a couple of hours.

"require formatting to access, which I assume is due to the partitioning mess" More likely incompatible format.
 
Why not just use system SATA on a borrowed PC to add the 2 old HDDs as slave drives? Copy to cloud or DVD or USB flash. Should only take a couple of hours.

"require formatting to access, which I assume is due to the partitioning mess" More likely incompatible format.

Thanks for replying! Unfortunately, no one who I'd feel comfortable borrowing from owns a PC, only laptops. I am an expat away from home and don't have a close network of friends.
 
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