I'm pretty sure it was the flash capacitor that blew up.
There's usually a voltage multiplier circuit that charges the capacitor. If one of those diodes failed, it could have reverse biased the capacitor. Electrolytic capacitors don't like reverse voltage. That causes the electrolyte to boil and... POP!
Smells awful and the smoke is not good for you, either.
I had a rectifier in a movie projector that blew the filter caps, once. They were as big as beer cans. When they went, they spewed foul smelling oil all over the inside of the housing. YUCK!
Given that I don't know the construction of the circuit that the flash unit uses, I could not advise on how to repair it or whether it is repairable at all. Generally speaking, the capacitor and the diodes that feed it could be replaced and it might work all right. However, that begs the question whether you have the wherewithal to do the job, not to mention the inclination.
It's probably best to junk the thing or scrap it for parts.
As others have said, flash capacitors can contain thousands of volts. Be wary of them! Be absolutely sure they are discharged (by shorting the terminals together) before you do anything in the vicinity of them.
When I was in Tech school, we used to cannibalize old flash units to make flash welders for soldering paperclips together.
Touch the wrong thing and you were in for a nasty jolt! If it goes through your chest it could quite easily stop your heart!