Can't turn on iPhone, no backup and updating isn't successful. Is there anyway to retrieve the data? I am posting this for a friend.  For some reason the iPhone 5 suddenly won't turn on anymore.  When the power button is pressed the screen lights up, the apple logo shows and then it switches to another screen where the following url is shown: support.apple.com/iphone/restore, the iTunes logo and the a picture of a cable beneath.

What I have tried:

I have visited the url and followed the steps listed on the page, namely holding the power and home buttons for ten seconds, waiting and then trying to turn the phone on again.

I have connected to iTunes and tried to update the phone but this is unsuccessful and produces an error, with error code 26 which gives me this link here: https://support.apple.com/en-gb/ht201210.  I have followed the steps on this page as well.

What I think the problem is:

I believe the phone started to update but didn't have enough space and at some point the update failed midway and now the phone can't turn on.  I believe this is also the reason why it wont update via iTunes.  I do not know however as I wasn't present during this time and my friend isn't sure either, all she knows is it wont turn on anymore.

The phone has no backup and my friend would very much like to keep / retrieve the data on the phone somehow.  Therefore we have avoided trying a factory reset at this point.  

We have taken it to a shop and they referred us to a specialist shop saying it would cost ~2000€ to decrypt and retrieve the data before performing a reset which is ridiculous.  

Are there any steps or methods we could further try before resorting to a factory reset and losing the data or having to cough up all this money?

Any help would be greatly appreciated.
Disk Drill or iFunbox may be able to help. It is evident that you are able to access the iPhone for a restore, so Disk Drill is likely the best choice. 

Disk drill will create the best backup that it can with the files left unencrypted, and iFunbox will only be able to access whatever the phone can.

Best of Luck!