ENGLAND, from Old English Englaland, "land of the Angles," from Engle, the Angles (< the shape of their original homeland, the Angul district of Schleswig)
The OED says of Angle
one of a LG. tribe that settled in Britain. XVIII. - L. Anglus, pl. Anglī, in Tacitus Angliī - Germ. *Aŋgli- (whence OE. Engle; cf. ENGLISH) the people of Angul district of Slesvig so called from its shape (mod. Angeln), the same word as ANGLE1.
ANGLE1 is "fishing hook".
The Proto-Indo-European root is *h₂enk- "to bend". The suffixed form *h₂enk-ula- (I assume with no stress on the initial syllable) became Proto-Germanic *anǥ-ul- becoming both angle "fishing hook, to fish" and Angle, England, English (AHD). Wordorigins has more.
The variant *h₂eng- became ankle from Old English anclēow and Old Norse *ankula "ankle", both from Proto-Germanic *ankulaz (AHD).
The other angle, "figure formed by two lines diverging from a common point," is from Latin angulus "angle, corner", also from *h₂eng-.
My question is: how did the Angles know their peninsula was shaped like a fish hook? Is it even shaped like a fish hook? (Angeln is to the right of the G of Schleswig.) There is an alternate theory: Angle might be from *h₂enǵh- "tight", as in German eng, Frisian ing "narrow" - since it is a peninsula. On the other hand, I'm not about to argue with the OED without more evidence.
Other cognates include Greek ἄγκυρα (as in anchor), and Old Church Slavonic ѫкотъ ǫkotǔ "hook", which uses one of those cool obsolete Cyrillic letters.