The double diode is likely a pair of schottky diodes for rectification coming out of the transformer. Anything with a reasonable voltage and current spec should be just fine.
However, checking these parts in circuit is simple. Just put on your diode check mode and measure across them. You should find a ~.6v drop on the transistors (both legs) and 0.3ish volts for the schottky diodes. If they're not shorted they're likely good.
Another thing; if it has no output, the problem is likely in the high voltage side (if the schottkys aren't shorted). It's exposed to the mains and therefore susceptible to spikes, etc. On a simple switchmode like this I'll run through the high side semiconductors with the diode check, which should take less than 5 minutes. Then I'll check the resistors starting with the higher wattage ones because they're more stressed. These check in circuit just fine; all your looking for is something close to their value. They fail open circuit, so anything less than their rating indicates they're good. If the fuse hasn't blown, likely the driver transistors are fine. I wouldn't even begin to bother with the low voltage side except to have a visual look at the capacitors.
When you find the problem and fix it, initially power it with a series light bulb to protect the new parts while you make sure it's now functioning normally. Often blown transformer driver transistors are a symptom of some other problem.
Replacing parts willie nilly is a waste of time. Find the problem and fix it.