Repaired water damaged iphone won't connect to laptop

I have an iphone that was sent through the wash. We sent it to an iphone repair shop, they fixed it so the iphone would work, but the screen is damaged so we'll need to get a new one. I'm trying to backup the photos to my laptop or iCloud, but iCloud says it doesn't have enough storage, and my laptop doesn't read the iphone as existing. I mainly want all the pictures and videos back. The phone DOES work as it will open, unlock, and run apps. What's the best way of getting back all my lost data?