You have the correct font, but Illustrator still isn't seeing it.
Being unable to open or close fonts from within Font Book's interface is almost always a sign its database is damaged. This also commonly causes issues such as you are seeing. The font is active, but not all apps are aware of its existence.
Restart your Mac and immediately hold down the Shift key when you hear the startup chime to boot into Safe Mode. Keep holding the Shift key until you see a progress bar towards the bottom of the screen. You can let go of the Shift key at that point. Yosemite is a bit different. Whether it's a Safe Mode boot or a normal one, you get the same progress bar. It just takes longer to get to the desktop in Safe Mode. So hold the Shift key until you get to the desktop.
OS X asks you to log in (you will get this screen on a Safe Mode boot even if your Mac is set to automatically log in). Let the Mac finish booting to the desktop and then restart normally. This will clear Font Book's database and the cache files of the user account you logged into in Safe Mode.