I had this problem a couple of days ago. Firefox is my browser. I fixed it by going to Tools/Clear Private Data, then unchecking ALL the options in the new window except 'cache', then clicking the 'clear private data now' button. After that, when I clicked on the Add Media option, the window came up in the middle of the screen, rather than being stuck halfway down. Then, to be safe, I also used Raincoaster's suggestion of grabbing the image URL from its listing in the Media Library (a tab option when you click on Add Media), going back to the post I was in the middle of writing, clicking on the little Tree image (for adding an image into the post), pasting the URL into that little box, and bingo, everything worked beautifully.
I am very un-techie. (I am sure you can all tell). But all the suggestions given by people on these forums worked for me, and were straightforward to follow. I didn't have to update any software or go via another site. Thanks to those experienced ones who stop the rest of us tearing our hair out!