OK, I purchased the same drive for my Toshiba Satellite tablet PC. I was able to get it up and running after an hour of searching around on the web. You didn't need to purchase any adapter, all you needed to do was take off the plastic casing (it is a square frame and goes around the optical drive), the plastic end cap, and the adapter that was on your original optical drive. You can normally purchase this setup online (it appears that you did) but there was no need to.
Anyway, you should take the battery out of your notebook first. The computer needs to power down and the battery may prevent full power down (mine didn't allow the drive flashing to finish).
Download Sony/Optiarc software located here. The drive comes pre formatted as a master and my Toshiba tablet didn't like this. I changed the drive to slave and my tablet didn't like that either. I suggest starting out with Reverse ATA, that one worked with my tablet. You may have to take the drive out of your computer, while it is off, and then put the drive back in your computer while after Windows fully boots. Windows will see the drive just fine and you can even use it. The problem is with your notebook's firmware which is having issues with the drive.
So wait for your computer to fully boot, insert the drive, run the software, change it to Reverse ATA, power down your computer (remember to remove the battery), boot your computer back up, and then Windows should reinstall the drive.
I have been able to burn DVD+RW, DVD-R, DVD+R DL, and CD-R discs. I even read a BD-RAM disc. So the drive works now, you just have to mess around with the ATA and/or Slave/Master settings.