If you can't get it clean, keep the frame and just replace the mirror.

Is it possible that you were fortunate enough to have purchased a very old mirror that just isn't going to respond to modern cleaners in the way you would like. I have a very old mirror that belonged to my grandparents. When I clean it, it does tend to look foggy, even though the actual mirror backing is lacking in only a few spots. The general overall effect is somewhat cloudy. I love it because of its family history and it's in our guest room where great mirror expectation isn't extremely important. Maybe you got something special and/or historical. If not, that's easy enough to determine. Replace the mirror and keep the frame as k9 suggested.
Annie1971

The problem may be with the reflective coating on the back. Try placing a piece of foil behind the areas that are cloudy with the shiny side of the foil facing the front and check to see if that improves the image.

Thanks for the ideas. I did remove the mirror from the frame to check the back but unfortunately that didnt help. Also, I dont think I can replace the mirror because of the shape and size. The marks are strange because you can only see them in certain light but when you do it makes the mirror look dirty. I imagine this is why someone donated it to goodwill. Thanks for your suggestions tho!

You can get a new mirror cut to what ever size you want at a glass shop, not as many around as there used to be. I had a friend who had an old mirror re-silvered at a glass shop once. It was antique, beveled with a cut design in the glass. It seemed to work well. I have no idea how much it costs for something like that however.