Check if the laptop can support SATA 3.0g/s

You need to a USB to SATA/IDE adapter.

You can find them cheaper than $20. Like this one on Amazon for $15. Here another USB to SATA/IDE one for $10.

I found a great video that explains the entire processing from connecting your hard drive to your laptop USB.

Based on the model number of HTS543232L9A300 on that label the drive is a Hitachi Travelstar 320GB 5400 RPM SATA 2.5" laptop hard drive.

But that is definitely not a standard SATA connector; I bet that thing connected to it is some kind of laptop connector interposer/bridge. I would see if you can pull that off because I am pretty confident that beneath it there is a standard SATA connector.

Just look at these images I snagged off of a similar model on eBay; clearly comes with standard SATA.

EDIT: Since the original poster says the connector is part of the main logic board on the drive, another idea would be to simply purchase another logic board for the same exact model but make sure it has SATA connectors. Mind you this should be considered a last resort effort since sometimes the logic board connected to a drive could be fine tuned specifically for that exact disk stack. But if you feel you have no other choice, get a logic board from a similar drive & then just swap that in to see what happens.