I would say Cook, he was their most influential and best player in their two biggest wins of the last 15 years - Australia 2010/2011 and India 2012. He has been a rock for them for nearly 12 years, in a position that has given them plenty of headaches over the years. He has had more than dozen different opening partners.
Some would argue Pietersen, since he was devastating and a game-changer, but in spite of his genius, England would not have won those two series without Cook's consistency.
Anderson runs him close though, he has been the leader of the attack since 2008 and has won England many games. He has been a more reliable consistent performer than Pietersen.
Cook
Anderson
Pietersen
Root/Broad
Stokes
However, if we go back 15 years, then Nasser Hussain, Vaughan and Flintoff deserve a mention. Both played a big role - especially Nasser Hussain - in the revival of English cricket in the early 2000s, after they hit rock bottom in the late 90's.
Vaughan took over from Nasser and carried on from there, his leadership was a big part of the success that England enjoyed in the 2000s.