It's "English" rather than "Anglish" because the vowel was subject to palatal umlaut. Umlaut is a process that occurred in many Germanic languages, Old English among them, and that caused vowels to change when they were followed by an [i] or [j]. The word English is from an Old English adjective Englisc derived from an Old English noun Engle that developed from an earlier form with i in the following syllable (Wiktionary gives the form as "Proto-West Germanic *angli").
Before a nasal consonant such as [n] or [ŋ], it was usual in Old English for a short a-sound to turn into [e] when affected by umlaut. (Meizi Piao. 2012. "An Analysis of I-Umlaut in Old English". SNU Working
Papers in English Linguistics and Language, page 84)
Other examples of nouns showing the same change of original short *a to to Old English [e] due to umlaut are bench, from Old English benċ, from Proto-Germanic *bankiz, and stench, from Old English stenċ, from Proto-Germanic *stankwiz.
In England and English, the Old English [e] sound was further changed to [ɪ] due to the influence of the following velar nasal [ŋ]. The same sound change occurred in string, from Old English streng, from Proto-Germanic *strangiz.
The variant vowels in Old English
The variants Ænglisc and Anglisc do exist in Old English, but I think they were less common than Englisc. They do not seem to have contributed to the ultimate outcome in Modern English.
I'm not entirely sure why alternative spellings with Æng-, Ang- and Ong- existed for this adjective in Old English, but the Oxford English Dictionary entry for the noun "Engle" has some discussion of its Old English byforms Angle and Ongle (Old English "a" often varied in spelling with "o" when it occurred in a stressed syllable before a nasal consonant such as /n/), saying that they "show alteration either after the place name (and combining form: see below) or after classical Latin Anglus Angle".
The note on the combining form reads:
Angel, the name of the continental home of the Angles, occurs as a place name in English contexts from Old English onwards (in Old English as Angel, Ongel, in Middle English as Angle; in modern use also commonly Angeln after the German form of the name). Frequently in Old English as a combining form, sometimes in the sense ‘of or relating to the Angles (either on the continent or in England)’, but more commonly in the sense ‘of or relating to the English people or England’ (compare senses 1 and 2), as Angelcyning king of the English, Angelcynn the English people, (rare) the Angles, Angelfolc the English people
It seems that this place name did not show umlaut, so the vowel in the ethnonym and adjective might have been occasionally modified by analogy with the place name.
The general history of Old English æ
If one of the Old English alternative forms Ænglisc or Anglisc had survived until the present day, it might well have developed into the form "Anglish". The reference in Robusto's answer to ash (æ or ae) changing into E is a bit inaccurate.
In general, the Old English short æ/ash vowel developed to modern English "a", as in ash itself (from Old English æsċe), or in Alvin from Ælfwyn. Other examples of short ash corresponding to modern English "a" (not "e") are hat from Old English hæt(t) and back from Old English bæc. I'm not sure whether I can find an example of this outcome of short æ before a nasal consonant, though, since it did not generally occur in that context.
The Old English long æ/ash vowel (written ǣ in modern texts) was spelled the same way as the previous vowel in Old English, but pronounced differently, and developed to modern English e or ea. That's what happened to the vowel in even(ing) mentioned in user2310967less's answer. But this isn't relevant to the word English since this word didn't have a long vowel in Old English.
The Latin digraph "ae" found in aether and aesthetic is commonly simplified to e, but it is not the same thing as Old English ash: the digraph "ae" in Latinate words represents an original diphthong in Latin or Greek and was used in spelling far after the English letter ash had ceased to exist. In the context of Latinate words, "æ" is not a letter of its own: it is just a ligature of the "a" and "e" letters.