[gw204]

You're close. Drill out the holes as you said until you get good, clean wood. I like to put a little countersink on the hole as well. Using a small paint brush/old toothbrush/etc. give the wood core and exposed glass a few coats of unthickened epoxy allowing the epoxy to soak into the wood between coats (I'm usually too impatient to wait more than a few minutes though). Then inject thickened epoxy into the hole with a small syringe and allow it to cure. Once cured, sand flush redrill your screw holes and seal with your favorite caulking.

In order to keep runs, drips and sanding to a minimum, once the hole is drilled I completely tape over it using blue painters tape. Then I cut a hole in the tape about 1/16" larger than the counter sink. After injecting the epoxy, I'll place another piece of tape over the hole and apply light pressure to it until it's almost flush with the surrounding surface. When it's cured, I just pull the top layer of tape off, sand it down down to the first layer of tape, pull that tape and then fine tune it to get rid of the tiny edge that might be left.