Never been to Ireland or Iceland so can't comment on that.
Not sure how long your stay in England will be but I do recommend you spend a day in Oxford which is the most beautiful and peaceful town I've been in. Not much to do there other than visiting Tolkien's grave and resting, but still a place you wouldn't want to miss.
There's tons of stuff to do in London and people to meet and to be honest I don't have anything out of the ordinary to recommend there, other than trying to find a hotel north of Hyde Park. Even though it's a little far away from the touristy things, it's a quite place, it has cheaper accomodation options, lots of transportation and if you feel like walking, you can cross the park each morning which is a great way to start your day
Also, Liverpool is a very boring city, only worth it for the Muscial background and if your a soccer fan that might be the place you want to go and watch a game, though it will be expensive.
And my out of the box recommendation is to ditch Ireland and visit Scotland, it's a magical place with beautiful landscapes and cool castles and monuments. They have great food, they have great scarfs and people are a lot lovelier than those pesky British guys, even lovelier than Rich and that's not easy. If for some reason you decide you want to go there, I recommend you rent a car and drive, you can get there in less than a day from London and you could make a quick stop in Lancaster which is another little but beautiful town near the border.