Wedge a small screwdriver tip in any place that will open far enough, and smack the remaining glued spots. If you try to pry it open, it will usually crack where you don't want it to. With a little practice, you can open one of those without leaving any marks on the outside. Now to the insides. Of course, there may be bad components, but we'll check the easy stuff first.
Most wall warts consist of a small transformer, several diodes, and one or more filter capacitors.
Voltage regulation is usually done in the powered device, not the wart. By the way, some warts are made with only a transformer inside the box. Obviously, that kind will only output an AC voltage. Rectifiers, filter capacitors and regulators will all be in the powered device. If the wire is broken inside the cable, it should make contact momentarily and the powered device will come to life.