Most Irish immigrants would have spoken English, though some probably spoke Gaelic.

Far more than 10% of Americans have English or British Ancestors. You are probably misled by a recent popular map showing the self identified majority ancestry of Americans by county.

This map had several problems:

It asked people to self identify. They often self identified as some more recent immigrant group ancestry though they also had some English or British ancestry.

It identified the majority self identified ancestry. In many of these cases British ancestry would have been the second ancestry.

Vast areas of the country (mostly in the South) self identified as "American" for ancestry. This was mostly people of largely British Ancestry.

Probably the majority of Americans have British Ancestry of some sort.

Total numbers of immigrants (slaves not counted in these statistics) to the United States have been about:

In the 17th century 175,000, mostly from Britain (mostly from England).

In the 18th century 400,000 to 1m, still mostly from Britain.

From 1836 - 1914 30m from Europe (mostly not from Britain, though circa 4m from Ireland)

In 1840, when the wave of non-British immigration really started, there were 17m people in the USA. 2.5m were slaves. So, 14.5m were of mostly British descent. Mostly all 17m were native English speakers (importation of slaves had been banned in 1807. Most slaves by 1840 were multiple generations removed from the original immigrants and were native English speakers by 1840).

Over the next 80 years, 30m immigrants arrived. However, the 17m who were already here had 3 more generations of children. By 1920, the population was 106m. This came from the 17m (and their descendants, and about 375,000 immigrants per year and their descendants. And of course, both groups intermarried with each other). Still, by 1920, the descendants of the original 17m probably made up more of the population by some considerable margin than the later immigrants and their descendants.

For example, from 1820 - 2005 (185 years) 4.8m people immigrated from Ireland to the USA. In 2008, 36m Americans self identified as having Irish ancestry. A lot of people have at least one Irish ancestor somewhere in their family tree after 7 generations of steady immigration. By the same ratio, if there were 14.5m people of mostly British descent in the USA in 1820, we would expect there to be at least several hundred million people in the USA who could self identify as having British ancestry (which would be most). (after all, those 14 million people were all here 7 generations ago, whereas the Irish arrived gradually and almost none had 7 generations of descendants in the USA.)

In 1910, 13.5m immigrants were living in the US (total population was 92m, so immigrants were almost 15% of the population, probably the highest percentage ever except for very early colonial times.)

From 1914 to 1954 probably negative immigration (1.5m Mexicans deported, very little immigration).

In 1970, Europeans were still 60% of the total foreign born population of the US, but the change of immigration policies in 1965 meant that by 2000 only 15% of the foreign born population was from Europe.

From 1965 legal immigration has averaged circa 500,000 people per year. On top of that, some 10-11 million illegal immigrants have entered the country in recent decades (some of whom have since become legal). (Though since the Great Recession in 2008, the number of illegal immigrants residing in the country has declined by about 1m as economics have caused some to exit.)

So, there have been circa 60m? immigrants to the USA in 400 years.

In 2010, the foreign born population of the US was circa 40m out of 310m total (13%)

The foreign born population of the USA has never been higher than 15% since early colonial times (and often much lower). Most of the population (though not all) that is not foreign born is English speaking (Usually fluently, if not as a birth tongue, if second generation. Usually their primary language if third generation). So the overwhelming majority of the population has always spoken English.

The percentage of total immigrants who spoke English, however, is much lower. If there have been circa 60m immigrants over the past 400 years, probably between 5% and 10% of them spoke English as their primary language.

It is the natural increase of the original English speaking immigrants (over many more generations than the natural increase of the later immigrants) which established English as the language, and then the adoption of English by the children of waves of non-English speaking immigrants together with the continuing natural increase of already English speaking populations, which has ensured that English continues to dominate.

Source for immigration statistics and population statistics, US census reports and http://en.wikipedia.org/wiki/Immigration_to_the_United_States