I would certainly put a custom L-shaped or square rug at the island. As for the kitchen sink, I would keep the rug as-is as the shape is suitable.

L shaped rug probably wouldn’t work as our island is more angled. See pic

Oh! I’m not expert but I thing a custom area rug or a round rug would be the best options here. Alternatively, you could run an ordinary rectangular rug from the island towards the sink.

Or even placing a rectangular rug between the sink and island would work great. Of course it wouldn’t fit perfectly, but sometimes that’s what a kitchen calls for; embrace the unordinary!