One study estimates the Anglo-Saxon contribution at 38% in contemporary East England. Another finds that the Roman and pre-Roman era East English were similar genetically to the modern Welsh population of England. The Anglo-Saxon contribution was closer to the modern day Dutch and Danish people.
This is not an easy thing to distinguish, however, because the populations of all of these regions have broad general similarities to each other.
Razib also covers these publications but with an important additional observation (emphasis in original):
So genetics tells us that extreme positions of total replacement or (near) total continuity are both false. Rather, the genetic landscape of modern England is a synthesis, with structure contingent upon geography. But, it also shows us that substantial demographic change which produces a genetic synthesis can result in a total cultural shift. Though we may think of elements of culture as entirely modular, with human ability to mix and match components as one might see fit, the reality is that often cultural identities and markers are given and taken as package deals. But, it probably took the transplantation of a total German culture through a mass folk movement to give the Saxons enough insulation from the local British substrate to allow them to expand so aggressively and become genetically assimilative and culturally transformative.Off topic, Razib also points in another post out an interesting effort to use folktales as a means of tracking Indo-European phylogenies, in much that same way that one would use lexical correspondences for the same purpose.