Photoshop :: Gradient Edge

I have a grayscale image that's a pretty consistent 85-90% black towards
the edges. I want to print it on a full page in a book, but I can't use
a bleed, so I want to fade the edges to white. I can only alter the
outside 1/4".

I figured out how to select a "frame" of color on the outside (select
the inside and Select > Inverse), but my attempts at using a gradient
fill on this frame have not been working too well. A radial gradient
fill (started from the center of the image) is too light by the time it
gets to the edge of the frame, and because the frame is rectangular,
it's not consistent. Linear gradient only seems to affect one edge of
the frame at a time.

What's the best way to fade my
1/4" wide frame from almost black to complete white, inside to outside?