Coded b1 as initial position, bn as position before n'th jump
+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| w | w | w |b23| w | w | w | b1| 6 | w | w |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | w |b24| 3 | w |b21| 6 | w | w | w | b2| w |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | w | w |b22| 10| w | w | 15| w | w | w | w |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w |b25| w | 9 | w | w |b20| w | w | w | 12| b3|
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | w | w | w |b19| w | w | w | 8 | b4| w | w |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | 11| w | w | w | 12|b18| w | w | w | w | w |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | w | w | w |b17| w | w | 10| b5| w | w | w |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | w |b16| w | w | w | w | w | w | w | 17| 13|
+---+---+---+---+---+---+---+---+---+---+---+---+
|b15| w | 9 | w | 9 | w | w |b10| w | b6| w | w |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | w | w |b13| w |b11| w | w | 7 | w | w | b7|
+---+---+---+---+---+---+---+---+---+---+---+---+
| w |b14| w | w | w | w | 11| w | b9| w | 11| w |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | 4 | w | w |b12| w | w | w | w | 7 | b8| w |
+---+---+---+---+---+---+---+---+---+---+---+---+
As for methodology, I started by deducing the following:
+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| | | | | | | | | 6 | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | | | 3 | | | 6 | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | | w | b | 10| | w | 15| w | w | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | | | 9 | | | | w | | | 12| |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | w | 8 | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | 11| | | | 12| | w | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | 10| | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | w | w | w | w | w | 17| 13|
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | 9 | | 9 | | | | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | | | 7 | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | | | 11| | | | 11| |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | 4 | | | | | | | | 7 | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
From there it seemed as if there were three points of high tension,
The top left 3, the bottom left 4, and the top right pair of sixes. I tried connecting the top left and top right a few different ways with trial and error, and came to the conclusion it was impossible.
From this conclusion I started working on solving the 6-area at the top right, because I had to work around a fair amount of filled squares, knowing I needed to head down rather than left.
Knowing there was an 'end' in the top right quadrant let me build a skeleton of knight moves out of that area, over to the 4, up around the pair of 9s, and almost connect back to the 3. I don't remember exactly the moves here, but I believe my chart looked something like this: (g being 'pencil' b)
+---+---+---+---+---+---+---+---+---+---+---+---+
| 11| | | | g | | | | g | 6 | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | | g | 3 | | g | 6 | | | | g | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | | w | b | 10| | w | 15| w | w | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| w | g | | 9 | | | g | w | | | 12| g |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | g | | | w | 8 | g | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | 11| | | | 12| | w | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | | g | | | 10| g | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | g | | | w | w | w | w | w | 17| 13|
+---+---+---+---+---+---+---+---+---+---+---+---+
| g | | 9 | | 9 | | | g | | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | | | g | | g | | | 7 | | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | g | | | | | 11| | | | 11| |
+---+---+---+---+---+---+---+---+---+---+---+---+
| | 4 | | | g | | | | | 7 | | |
+---+---+---+---+---+---+---+---+---+---+---+---+
Doing some counting, I was clearly missing something in the bottom right, fortunately I saw a cycle which could be safely added.
+---+---+---+---+---+
| g | | g | | |
+---+---+---+---+---+
| | 7 | | | g |
+---+---+---+---+---+
| | g | | 11| |
+---+---+---+---+---+
| | | 7 | g | |
+---+---+---+---+---+
A little more counting let me verify that these were basically right, shade them to black and white, and put the last one to the right of the central 12 to link the top and bottom halves of the left side leading to the graph at the top of this post.