Welcome to the forum.
Did this get fixed? If the unit won’t even turn on, there’s not much help that we can give other than to suggest you take it in for service.
Also, make sure that the problem isn’t with the power outlet. Did you try plugging it in elsewhere?
As for finalizing the DVDs, you’re going to have to get the unit working again in order to do that.
As an absolute last resort, you can do some research on programs called ISO Buster and ISO Puzzle which will allow you to extract the useable data from those discs, but whether the video will turn out correctly is hard to say.