Well first of all you tried and failed. That's what The War of Independence was.
As for why they didn't commit more resources and men to the fight, you have to seriously ask yourself whether going complete World War on Ireland would have fixed anything, rather than made it even more radical. It was their extreme brutality and lack of mercy in 1916 that radicalized the population to the point that large scale, countrywide rebellion became inevitable. They population got even more radical during the conflict due to rampant British war crimes. At every point, greater British brutality and military commitment increased nationalist sentiment.
Regardless of what Britain did, I find it hard to envisage a timeline where they could have maintained control of the modern Republic up to the present day. From that perspective, they were right to just back out after a few years of fighting. From any moral perspective they shouldn't have been there in the first place.
Also, that picture's 1916, not the war of Independence. They did win that, then fucked up beyond belief in the aftermath.