The first step is redrawing the board so that the unintuitive knights moves are replaced with something more visible. Like this, where the connection is made by adjacency:
Here, we start with black tokens on the black squares and white tokens on the white squares, and we must switch them.
Trial and error reveals the following thirteen move solution: FB, DF, AH, IA, GD, FG, BI, AB, CF, DC, EA, BF, HD.
Notes: I solved this in about five minutes. I managed to find a thirteen move solution the first time I tried it, but when I went to write it down, I could only find fourteen move solutions. Aargh!