+10 for madcratebuilder

Go with the "Super Glue" or cyanoacrylate glue that madcratebuilder recommended, Balsa Wood Model Aircraft (Radio Controlled Aircraft) use it to glue the balsa model together and you will not find a better method.

The super glue will flow and be absorbed by the surrounding wood on either side of the crack or joint and literally be "welded" together.

Below, balsa wood RC Aircraft wing with landing gear strut being super glued in place.



Three different viscosities.