i Had the same problem with unresponsive screen, stuck on the password lock screen, and when i plug it to my pc the i tunes doesnt recognize the device and refuse to back it up.
However i solved it by going to itunes accounts menu and pressing on (authorization), it will ask you to provide apple account information, and once that's done your iphone is now recognized and you can back it up.
i Also used another softwares called iexplorer which allowed me to convert the encrypted iphone backup information to actual files (photos, videos....etc).
i Am glad I was able to solove this, as i spent two hours waiting in the apple store just for them to tell me they can only resortore the device and all my data would be gone.
hope this helps