The Empire as described in "Mirror, Mirror" is, in the best(?) traditions of SF, something of a straw man. It is an extreme version, almost a parody, of a working empire. Promotion by assassination? Barely controlled aggression among the ranks? Torture as a normal disciplinary action? These are not the hallmarks of a stable social order, and Kirk's prediction that the Empire will not last is probably correct. So, the OP conflates two different questions. "...why does it fall after becoming peaceful? Why does peace and reform necessarily result in weakness?" is two entirely separate questions.
First, "...why does it fall after becoming peaceful?" is difficult to answer since the succession society is not described in any detail. Given Roddenberry's utopian tendencies, the TOS version became pacifist and demilitarized. While this has its attractions, such a society is unlikely to successfully resist an aggressive, effective military force. It's a truth which many folks don't like to admit, but in the short term, unopposed use of force works just fine.
The second question, "Why does peace and reform necessarily result in weakness?" is easily answered: it doesn't, as long as it does not destroy the ability to engage in war. Vegetius's "Si vis pacem, para bellum" (if you would know peace, study war) goes back a long ways, and stands in stark contrast to J.B.Miles' assertion "If nations learn war and prepare for war, they will have war. If they study peace and prepare for peace, they will have peace". Both are, in my opinion, equally wrong. It is the middle way, of maintaining capability while avoiding the exercise of that capability, which produces a workable situation.
A demilitarized society trying to counter a determined enemy will encounter a hard truth: the military is different from civilian society, and it cannot be built quickly. Warships are built to an entirely different set of standards than freighters, and effective ones need to be built as such from the keel up. Warships carry no useful cargo, so they are completely uneconomical. They are built to unreasonable expectations of speed, armor and armament, and none of those things are useful in peacetime. The same goes for military equipment of all sorts. Fighters and bombers are useless in peacetime. Artillery, and all sort of weaponry, make no sense at all in the absence of war. But they are fundamentally required by a society which is opposed by a society which possesses them, and willing to use those tools to destroy or subjugate that society.
Much worse is the fact that military society is fundamentally different from a peaceful, industrialized civilian society. Where civilians value cooperation and affability, the military, in order to be effective at what defines it (combat) must encourage (and simultaneously control) aggression and conflict. Where "Star Trek" civilian societies hold individual lives precious, military forces must be willing to sacrifice any number of individuals and keep on functioning (See TNG, "Thine Own Self"). Where societies which we admire protect individual autonomy, military organizations must enforce a level of discipline which will permit some to sacrifice others for the good of the whole - whether the ones being sacrificed agree or not. These patterns of thought, and feeling, and behavior do not come easily to many, while they come too easily to those who would abuse them, and the development of a working military culture is not something which will happen quickly.
Such aspects of military behavior are easy to parody, as Roddenberry did in "Mirror, Mirror", but accepting such parodies as essential truths is a simplistic, willful ignorance of the logic of conflict.
