I have the exact same issue, the USB port is defective because my USB sometimes works as well. I've had my new CLA 250 in for a service already to get this fixed, they've replaced something and it still doesn't work.
I had big problems with this when I first got my CLA. My iPhone 5 kept randomly disconnecting from CarPlay. Tried my wife's iPhone 5 and still didn't work. Tried using a friends iPhone 6 and it worked fine. Got an iPhone 6 that day and haven't had a problem with it. I read on Apple forums that people had this problem and a lot put it down to a dirty or damaged charging port on the iPhone.
Just bought by CLA yesterday. CarPlay has been working fine until this morning. The port seems to be working (i.e. it charges and you can play iTunes through car media) but it is saying CarPlay capable devices not available. I tried using different cable without success. Restarted my iphone and restarted the car. This is pretty frustrating since I just got it and am already having issues.