if you replaced the screen and the device went off, then it could be the screen which sucks to much energy and that could damage your iphone. plug the screen off and restore with that opened iphone. if that doesnt help, i hope you will only need a new battery.
the online way to backup your files is with a working screen. order one and after work send it back if you dont need it. to open an i9100 or s2 you just watch that very easy youtube videos. its very easy, be careful. if you dont want to order a sceen, you will have to load clockworkmod on your s2 and get connection over ADB or other tools. maybe moborobo or http://developer.android.com/tools/help/... there are different ways to backup your files. over cwm you can start your device in usb mode and copy a usual from a usb drive. and over adb you have command line, which is more complicated but possible as i know.