You are perhaps confused by the name Irish Republican Army. A "proper military campaign" would have required the IRA to be holding and defending territory, so that the British could wage war against it, destroy it in a campaign- that's what you would expect of something called an army. The IRA was a guerrilla movement, an insurgency. It usually carried out constant but sporadic attacks against weak military targets. Small bands, "flying columns", would attack isolated targets like police barracks and then disappear . When the Irish attempted to act like a real army and hold territory ( like the early Easter Uprising) they were pretty quickly defeated.

A better question would be why the British did not mount a bigger and better counter-insurgency operation. A guerrilla movement depends on being able to hide among the population, and a great problem therefore is that the civilians bear the reprisals for whatever it does. The British certainly engaged in reprisals. They blew up houses of IRA members, and created a force of "Auxies" and "Black and Tans" or special constables who acted as paramilitary units, often without direct control. These paramilitary soldiers ( often ex-soldiers from WWI and sometimes also Irish) were willing and able to use violence to terrorize the population- on one occasion, they destroyed the center of Cork in reprisal for an attack on one of their units- but the British government was divided on how much force should be used. If it had become more Cromwellian and bloody-minded, perhaps it could have eventually defeated the IRA but the force the British did apply just widened Irish resistance- in 1921 dockworkers and transport workers went on strike, refusing to ship munitions or transport soldiers, and the entire court system broke down. As the conflict widened the expense of the conflict grew as well- and this was only a few years after the exhausting WWI. If some British didn't mind if innocent civilians were killed and Irish cities burnt to the ground, few felt there was money to pay for it. At about the same time the British realized negotiations were unavoidable, fortunately most of the IRA did also; importantly Michael Collins, who led the military effort. It should be noted that the settlement with the British did not end the troubles- there would be several years of continuing conflicts within the IRA, and of course later in Northern Ireland.

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Charles Townsend: The British Campaign in Ireland, 1919-1921

Peter Hart's more recent The IRA at War, 1916-1923 also looks to be excellent- but I have not read it.