Listen as I don't want to waste your money. However here is the thing.
If the hard drive is mechanically sound, I mean hardware is totally okay. Then yes, absolutely the adapter will happily work. You have to plug in the laptop hard drive into the adapter. It makes the hard drive like a big usb thumbnail drive.
If the hard drive is fine, Windows on your gateway will install the usb controller and you should be able to see the hard drive in My computer.
I have done this solution MANY times for people who have there laptop get destroyed when they spill coffee on the keyboard.
Please be careful as the laptop hard drive does have moving parts, so set up the laptop on something like a book close to the pc to ensure it doesn't move around. Once it is going, you will be able to easily copy/paste your files over.
I have to say this again, the adapter only works pending the hard drive is mechanically sound.