Can't get my Iphone 5S out of recovery mode?

So two days ago I tried to download the newest Apple software update (7.0.5 I think!) onto my new IPhone 5S, my phone proceeded to switch itself off (which is usually the norm when downloading an update) but when it turned back on it just had a black screen with prompts for me to plug it into itunes - this is all my phone has done for two days and I can't use it for anything even after trying a hard reset etc it still won't turn on properly.

When I did plug it in, itunes told me that the phone was in 'recovery mode' and that I'd need to restore it from backup in order to get it to work. When I tried to do this it wouldn't allow me and instead prompted me to download the new software through itunes onto my phone which would get the phone out of recovery mode.

I've tried to do this twice now and after waiting for two hours (each time) for the updates to download, it finishes and then itunes says that it's 'processing the update' but both times I've been greeted with 'error 9066' telling me it's unable to complete the process as I'm not connected to the internet - which I clearly am as I have full Wi-Fi signal and everything is operating as normal on my laptop.

Can anyone help me save my phone without me having to send it away and lose all my photos and messages etc?!

Did you jailbreak? Because if you did that would explain the issue.

The next thing I can think of is that the version of iTunes that you are using is either outdated or corrupt. I would recommend totally uninstalling iTunes and all of the associated programs.

Uninstall iTunes, QuickTime, Apple software update, Apple mobile device support, Bonjour and Apple application support. Then reinstall iTunes and try again. If that does not work try on another computer. If that does not work then the hardware of your phone is damaged.

Put it into dfu mode.
and then restore through itunes.

if that doesn't solve the problem uninstall itunes restart computer. Download latest itunes from the apple website.