I love that hole in the side of my ukulele. I insist on putting them in every uke I make. It sends the sound up to me in a crowded noisy jam, and it does not hurt the overall sound of the uke at all- in fact it might help it.
But it is obvious, it must weaken the side of the instrument, so I glue a veneer patch on the side with the grain going opposite the grain of the side. Here I have a caul and clamps holding that patch tight as the glue dries.
Glue dried, Patches on, Ready to drill the hole out.
I rough the hole out using a drill bit in a Dremel tool. Then I switch to the Dremel's sanding disk to slowly and carefully finish the hole and fit the wooden oval ring into it.
Now that it fits into the hole, I carve it down flat with a very sharp chisel and apply superglue to cement it in.
Then it is just a matter of sanding it down flush.