It's hard to give you definitive advice without knowing a bit more, but I'm assuming it's a laptop (hence hinge broken!), and that the backup is on the same hard drive as the Windows installation. If it's in fact an external USB backup drive then it's much simpler, but if it's internal and on the same drive as Windows, I suggest that - as someone who doesn't know much about partitions etc. - you will find it easiest to open a live USB session of Ubuntu, then copy your files from the backup (or you may even find the originals are accessible) to a second external drive.
I'm not sure how far you got ... it looks like you may have tried to install already, so you may have an Ubuntu iso USB already but instructions below if not.
(1) You will need a Ubuntu installation USB ISO - if you don't have that already you'll need to get access to a working computer and a USB stick with at least 4GB capacity. Anything on that USB will be deleted in the process. Go to https://ubuntu.com/download/desktop to download the file, and go to https://ubuntu.com/tutorials/install-ubuntu-desktop#1-overview to see simple instructions for how to do it.
(2) You then go to your unbootable computer, put the USB in and turn it on and hit one of the keys that allows you to enter the boot menu. Different manufacturers use different keys to do this - it's likely to be one of f2 / F10 / f12 / Esc / Del. The correct button to press will likely flash on the screen very briefly before it tries to boot up. Once you are in there, you should find a setting allowing you to select boot order. Make sure USB is the first option. Save the changes and exit, then reboot.
(3) If this goes well, you should then get to the Ubuntu installation setup screen where you are offered the options of installation or 'try Ubuntu'. Do not choose to install Ubuntu (you risk losing the data you want to keep). If you select 'try Ubuntu', it will open a fully functioning version of ubuntu but do nothing to your hard drive with all the files on it that you want to keep. Once it's fully loaded, you should be able to find your hard drive and locate the things you want to save. You can then copy then to a second USB stick.
(4) Once you've retrieved it all, you can go back to the installer, reformat the whole drive and start again from scratch.
There are undoubtedly better ways to do this, but they carry some risk of data loss if you don't know what you're doing, and I think for someone that is unsure about partitions etc, this is probably a pretty safe thing to try.
