What’s the best way to repair the cabinet wood?

I just finished working on a neighbors table where one of the 6 legs broke off.  Purchased from the hardware store a small bottle of Gorilla glue, it came in a 4 oz. bottle and is clear in color.  Pretty simple to apply, using a spray bottle mist one side of the gluing surface with water, the other side use a small amount of the Gorilla glue and clamp.  Leg is very sturdy and table (8 ft. oak) is rock solid.  That should work for you w/o fussing with nails.

My experience with Gorilla glue is that it foams and expands.  It needs moisture in order to he activated -- hence spraying one side with water.  They may have reformulated it since I used it.  The critical word in Jato's comment is "clamp".  Figure out how you are going to clamp your repair before re-assembling the repair.  Many woodworkers use brads (tiny finishing nails) as their clamps.

It sounds like the glue that jato referred to is a cyanoacrylate glue with an accelerator spray.  Gorilla glues come in many, many different types, such as the amber colored foamy stuff which is polyurethane based, and their yellowish wood glue.  The number of glues on the market is mind boggling and the right glue depends on the specific application.  Right now we're using Gorilla Max Strength Clear Construction Adhesive to glue cabinet end panels and door casings in place where we can't nail them.  The glue came with the cabinet installation package from the cabinet supplier.  It comes in a caulking sized tube and is takes about 30 minutes to solidify.  The problem I've had with cyanoacrylate glues is that the bond is not very strong and it will separate if it's stressed.  

jato, is this a CA glue or some other Gorilla product?  My experience with CA glues is that they make a strong bond with tensile strength, but if the two pieces get a sideways impact or stress they separate relatively easily.  

Go to a local small hardware store and ask which glue they sell that would be best for that application.  Personally, if it's real wood being attached to real wood, I'd use the yellow wood glue, such as Gorilla Wood Glue, and apply pressure on it, as mentioned above, until it dries.  The construction glue that I mentioned would also work great, but it's messy to use and has to be cleaned up with denatured alcohol.  I'll be applying more of that today to finish off the kitchen cabinets I've been installing.  I'll use a pneumatic pin nailer to hold the cabinet toe space covers and corner trim in place while the glue sets.