Iphone 6 storage full message when it is not!

My wife 16GB Iphone 6 Plus gives a false message that its storage is full when it should be at least 12 GB free after deleting apps, messages and hundreds of pictures. It looks like the iPhone storage usage increases the more pictures and apps deleted. After deleting 5 GB photos and videos the storage decreased instead of giving 5 GB free storage. In settings - storage handling there are no more items that use much storage. Unfortunately this iPhone was never synced with iTunes and now it is impossible to sync because iTunes interrupts the sync with a message that iPhone storage is full. It is not possible to connect it to iTunes. But it is possible to connect it with iTools 3 party software for OS X and I can observe over 9 GB!!! is used by Other ??

What is Other? How can I access and delete the data iPhone keeps hidden somewhere? Because its not possible to backup it is no option to restore her phone to factory settings and loose all pictures and messages. Is it a faulty flash memory in this 16GB iPhone?