Too bad I didn't check this post while trying to solve it because gdsports got the solution! Thank you very much!
So it is indeed necessary to solder the pins. We didn't thought about this because we connected VCC and GND without soldering as well and the LED was working fine whereby we assumed that it should be enough to stick them together for now. Seems like we were wrong and I've got now three working gyros at home.
Again thanks to everyone for your help!