There's very little direct info on the Terran Empire's birth in canon. I'd lean towards the theory that the timelines diverged long before First Contact, and that the Empire may have its roots in the Roman Empire (Terran is derived from the Latin word for Earth, Kirk given the title "Caesar," etc.). So we don't know what technological differences there were between the prime and mirror timelines by the point of First Contact.
The concept of a mirror universe in general, and especially its depiction in ST, makes very little sense, but if we suspend our disbelief a bit, a case might be made that:
- The Terran Empire didn't immediately launch an attack on Vulcan with just the Phoenix and 1 commandeered Vulcan science vessel.
- Because they saw First Contact as a prelude to a Vulcan invasion, they engaged in a period of rapid technological R&D, just as the U.S. did during the Cold War, when they perceived the USSR as an existential threat. Except, the Terran Empire had the additional benefit of reverse-engineering Vulcan technology.
Given these two things, it's not inconceivable that in 92 years, the Terran Empire could catch up to the Vulcans enough militarily to conquer Vulcan. After all, it took Japan only about 50 years between being an insular society utterly helpless against European military technology, to developing a modern military and actually defeating the Russian Navy and Army in the Russo-Japanese War.
Also, remember, the time it took for United Earth to catch up to the Vulcans was prolonged as the Vulcans intentionally held back technology, wanting to give humans time to advance culturally before becoming a space-traveling power.
Additionally, if the T.E. had come in contact with other space-faring races, similarly assimilating and reverse-engineering their technologies (as well as conquering militarily weaker species), before they attempted to take Vulcan, that would further increase their odds of success. Consider the "Samaritan Snare" episode with the Pakleds, except the T.E. weren't just acting as technology-stealing pirates but were actually interested in understanding and improving upon the technologies they captured. 90-some years of such a singular global agenda could produce very profound results.