First things first: As I understand it, you want to put all your iTunes music and your photos onto your new 128Gb JetDrive rather than store them on your computer. Yes, you can do this, and it should work just fine.
To your computer, your JetDrive is simply an additional drive in addition to what is already inside your computer. I suspect your JetDrive is either defective, or it wasn't formatted properly.
I suggest you begin by erasing/reformatting your JetDrive using Disk Utility. (All the data on your JetDrive will be deleted, so plan accordingly.) If you aren't familiar with Disk Utility, be extremely careful—you can easily do a whole lot of damage to the data on your computer using this utility. I've never used a JetDrive myself, but it should have come with detailed instructions on how to do this. If not, or if the instructions aren't clear to you, I recommend you get someone with OS X experience to personally help you do it step-by-step.
If after erasing your JetDrive, it still takes hours to transfer your data, you likely have a defective JetDrive.
As for how to get iTunes to use your new JetDrive after successfully copying the iTunes Media Folder to it, you set iTunes to use your new JetDrive & iTunes Media Folder Location instead of the one on your computer. This option is located in Advanced Preferences of iTunes, in iTunes>Preferences...>Advanced. There you will see the current iTunes Media Folder Location and a button to change it.
Your photos work in a manner similar to iTunes, but instead of using iTunes Media Folder for its data, Photos uses Photos Library.photoslibrary. Unfortunately, getting Photos to use a new library isn't as straightforward as is iTunes. After copying Photos Library.photoslibrary to your new JetDrive, you must then either delete or rename the old library. Doing so will cause Photos to prompt you for the name/location of the library when it starts (since the one it is supposed to use cannot be found because you deleted or renamed it.)
Lastly, after you have successfully put your music and photos on your new JetDrive, be sure Time Machine is set to back it up from there. You do back up your data, right?