An intriguing question was asked by Arthur and it is one which I would never thought to have asked. My first thought is that it might have been how it was spoken rather than written. So, I quickly checked to see whether any similar words starting E- in old English were now said starting with an I-. The main types of word in this area of "End" words of various forms and none of them have changed so a change in the way we write E- doesn't seem to explain it.

So, next I had a look to see what In- Ing- type words meant, and I find

In (in) and
Ingan (to go in)

At this point I'm tempted to suggest that "Ingas" are a pun on "incomers". But then I checked to see what "Eng-" means, and we have

"Engel" (angel)

Remember the pun by Pope Gregory I in latin: "Non Angli, sed angeli" (They are not Angles, but angels) said around 600AD reported by Bede to have been Gregories words when he first encountered pale-skinned English boys at a slave market, sparking his dispatch of St. Augustine of Canterbury to England to convert the English.

Looking around again I find :

Aenglisc = English

but we also have

aen (one)
aenan (unite)
aengancundes (?in a unique way?)
Angel = English
Angle ('Angle' as in a corner)

Aen, suggests the possibility of "union" so the "Aengan", might be translated as "The Union". However, the ending -isc ('-ish') intrigues me so I have a look at other words for peoples in Old English to see how common it is. I find

"Wealh" for "Welsh", but also "Wilisc" (Welsh)

Looking at "Pict" I find:

Pehta, Pihtas, Pyhtas, Peohtas (Picts)

We know the Picts vanished into history well before 900AD (<750AD?) so these forms are relatively early.This seems to suggest the -isc suffix is relatively late and that "wealh" is an original version which has a later suffix of -isc, '-ish' added perhaps after the arrival of a new influx of Anglo-Saxons?

On the assumption that the original suffix was "-as", this would suggest the Old English (pre Anglo-Saxon) version might have been "Ingas" (incomers) ... which is not that dissimilar in meaning to "Wealh" (welsh) but also meaning "foreigner".

So perhaps we say "Inglish" as a pun on those European Immigrants or "Ingas"?