I know that you can re-felt. I have done that with good success. Just remember that it will shrink more (nature of the beast) so watch it.
i don't really know how you can patch that hole though.
I guess it depends on the size of the hole, but that's what i would do. I would do is string yarn across it in about 4 to 6 places (like a spiderweb) before refelting it and just hope for the best.
If the hole is larger, you can knit up a swatch and felt it to match how much the sides of the bag are currently felted. Cut it to fit inside the hole, and then stitch it into place with more of the same yarn and then refelt.
I don't think either of my ideas will be perfect though.
Best of Luck!!!