Power IC problem, Computer wont recognize, How to backup phone?

Hi Guys,

My iPhone 6 refused to charge yesterday and so i brought it to a Service Center (because no apple care) and they said it was a problem with the Power IC and that they couldn't fix it because they didn't have the parts.

So I am planning on buying a new 6S today to replace my phone but my problem is i haven't backed up my phone recently (October was the last backup, stupid I know) and my computers wont even recognize the phone. I have tried to do an iCloud backup but it gets stuck on "estimating time remaining" for about an hour with no bar movement. So my question, is how can I backup my phone in its current form or how what would be the best way to get the data off the phone?

Thanks so much