The irony of war
Poland was one of the first nations to be attacked by Nazi Germany in World War II, so it came with ironic surprise. Also, it is ironic that Eva Hoffman should have to suffer the ultimate instance of 20th century warfare, at such a young, vulnerable, innocent age. The irony continues, because the War is also more horrifying and tragic because of the powerful technology that Germany developed, all of which was hidden in dramatic irony.
The irony of exile
Exile is an ironic problem for a young, innocent girl. Typically, exile is a mark of punishment, but this girl is punished for having done nothing wrong. In fact, she is exiled from her beautiful, peaceful home by violent, wrathful soldiers who were invading for the worst reasons: Power, empire, and world domination. Her exile is surprising to her, but it is also just ironic, because she has to suffer so much because of other people's malice and evil.
The irony of language
When Eva moves to a new home, she realizes that her entire identity will need adjusting. The sounds she makes will change. The sounds she understands will change. She has to learn English for survival and to help her family adjust to life as a refugee immigrant. In Canada, Polish isn't a helpful language, so ironically, she loses even her mother tongue in some ways. Her suffering is thorough and completely jarring.
The irony of home
When she finally finds a place to settle down, in the great city of Houston, Texas, Eva realizes that she is slowly beginning to feel Americanized. She decides to accept herself. She decides to belong to this culture, although ironically, it isn't remotely similar to what she used to enjoy. Once upon a time, she lived in scenic, rural Europe with a small community who were tight knit. So, home in Houston is ironic, because the city is huge, and her community is more anonymous than ever.
The irony of excellence
When Hoffman applies to go to Harvard, she is accepted. That means that the suffering was also accomplishing an unintended, secondary effect that she hadn't considered. Having suffered for so long, through so much frustration and confusion, she has become excellent and powerful. Something about adaptation makes her an excellent candidate for the highest of higher education, because her suffering has catalyzed ethical growth and power in her private identity.