first, you should take a voltmeter and check the voltage at different point.
Is it a DC/DC power supply or AC/DC ? If it is AC to DC, there should be a bridge, and may be a transformer. Some transformer have built in fuse..., you should check the voltage at the input and output of the transformer.
In switching power supply, there are chemical capacitor, which could die from temperature / or aging... you should check the voltage at the capacitor.
Then there are the switching component.
Last, but not least, there is the control loop, but it is the part that is the least likely to burn out...
Good luck.
PS : be VERY careful , you could kill you....