iPad Air 2 not recognized by iTunes

Hi guys.

I bought an iPad Air 2 64GB WiFi+Cellular the day before yesterday. So far, I haven't been able to get it to show up in iTunes. Neither on my laptop (Sony Vaio S, 2011) nor on my workstation (custom built, brand new). Both machines run Windows 7 Home 64 bit. iTunes was not installed on my workstation until yesterday, so it's the latest version. I also updated it on the laptop, and later uninstalled every bit of Apple software and did a clean re-install.

The iPad shows both under "Devices and Printers" and the "Device Manager" as "Apple Mobile Device USB Driver". When I connect it, I get the "Trust this Computer" message - every time. I hit "Trust", the prompt disappears, and nothing else happens. Next time I plug it in, the prompt reappears. Whatever I do, the device doesn't show in iTunes, the diagnostics say "No iPhone, iPod or iPad found". (Sorry for the screenshots all being in German, but you'll be able to understand them, I think ;-) )

I re-installed the drivers repeatedly. I "untrusted" all computers by resetting the security settings on the iPad. None of the official support threads seem to apply, because they all seem to be dealing with the device not being recognized by Windows itself, or a faulty drivers installation, none of which is (or seems to be) the case here. I restarted the tablet while it was connected to the PC. I tried every single USB port I was able to find. :-) This is my first Apple device (not counting a 2nd gen iPod nano, which connects fine, by the way) and it's not off to a good start, I'm afraid.

Any ideas? Thanks.

Jules