I've used LED tape to illuminate under kitchen cabinet uppers. It does work well there. But it is inherently straight. It wouldn't form into a nice circle shape, which is kinda what the shape is under a router body. But you could use short segments to form a rectangularish shape, or maybe a hexagon. That *should* work.
Tapes have minimum cut lengths, so you can't just cut them into arbitrarily short lengths. The one I have at hand has cut markings at roughly six inch intervals. I think the intervals encompass three LEDs, so if you want short segments you should buy high LED-per-foot tape.