Even if your screen doesn't turn on, but the tablet itself does, you should be able to move data from the device over to a computer. Then you can move it over to another tablet. Just connect it to the computer via USB. If the device has been used on the computer before, it should recognize it.
The only way to check the screen to see if the problem is fixable (without replacing anything) would be to open it up and see if anything is loose or disconnected. Yes, I know opening a tablet is never a simple task, but it can be done. There are tons of instructional videos online. Even ones on how to replace the screen. However, that said, I would not open it up unless it is no longer under any kind of warranty, as that would void that...