Go to your nearby Ace hardware store and tell them the story. They have a couple of products that work well. One is a liquid that hardens the existing mushy wood (after taking out all you can) and the other is an epoxy wood filler. I like the Wood Epox which comes in two containers that you mix together. It doesn't run like others do. It takes awhile to set up (several hours) but it can be drilled, screwed, etc. when hard. Use two disposable gloves and dip into each container with separate gloves. Do not use the same glove for each container which might contaminate the supply.
