The OED lists the etymology of height as follows:
Old English híehþo (also later héahþu) = Old Low German *hôhitha (Middle Dutch hogede, hochte, hoochte, Dutch hoogte, Middle Low German hogede, Low German högte), Old High German hôhida (Middle High German hoehede), Gothic hauhiþa, < hauh-high n.1 + abstr. ending -iþa: see -th suffix1. From the 13th cent. the final -th after -ȝ, -gh varied with t (compare drought, drouth). In Middle English the forms in -t were predominant in the north, and since 1500 have increasingly prevailed in the literary language; though heighth, highth were abundant in southern writers till the 18th cent., and are still affected by some.
The stem-vowel has generally been ē, ey, ei, though forms in i occur from 13th cent., especially in northern writers, hicht being the typical Scottish form from 14th cent.; in English hight is found from 15th cent., and was very common in 16th and 17th cents.; highth was also very common in 17th cent. and was the form used by Milton. The hei- forms come lineally down from Old English (Anglian héhþo); the hi- forms are due in the main to later assimilation to high n.1 Current usage is a compromise, retaining the spelling height (which has been by far the most frequent written form since 1500), with the pronunciation of hight.
For high, they write:
Common Germanic: Old English héah, héa-, héag- = Old Frisian hâch, hâg (West Frisian haeg, heag, heeg), Old Dutch hôh (Middle Dutch hooch, hog-e, Dutch hoog), Old Saxon hôh (Middle Low German hoch, hog-e, ho, Low German hoog), Old High German hôh (Middle High German, modern German hoch), Old Norse há-r (earlier hǫ́-r < *hauhar), (Swedish hög, Danish høi), Gothic hauh-s < Germanic *hauho-z < pre-Germanic *koukos: compare Lithuanian kaukas swelling, boil, kaukaras height, hill. Old English héah, héh, regularly gave Middle English hēgh, heygh /heːx/ , whence later hee (still in Scottish); but in 14th cent. this was narrowed to hiȝ, high /hiːx/ , whence hie, hy: compare the parallel phonetic history of die v.1, eye n.1
As with these words, Chaucer used both heigh (hey) rhyming with seigh saw, and hy, hye rhyming with Emelye, etc. The final guttural began to be lost in the 14th cent., as shown by the spellings he, hee, hey, hi, hii, hy(e; modern English retains the late Middle English spelling high, with the pronunciation /haɪ/ .
So height is spelled as a compromise, maintaining the pronunciation of "hight" while being spelled with ei to reflect the Old English ties. The ei form is older--as the OED notes, hight was created in later assimilation with the word high. High, on the other hand, maintains its Middle English roots. These examples show that when words are adopted into English, there are no set rules to standardize them. Height could have been hight (as it was, popularly, in the 17th century) but the Old English ei spelling won out.