If it came with a hard drive, that generation laptop I'm sure would have come with a 9.5mm HDD. You can replace it with any 7mm (might need a spacer/adapter) or 9.5mm SATA SSD. On an old XPS I installed a 7mm SSD into the bay that originally held a 9.5mm HDD. While I can add the 2.5mm spacer/adapter that Crucial included with its SSD, I don't think it's required or would add any benefit. The original drive bracket holds the new SSD solidly and securely once all four screws are tightened.
What size HDD is needed for an E5570 laptop? I know it's supposed to be 2.5 inches, but 7mm or 9mm?
What size HDD is required for the Latitude E5570?