inlet n.

a Middle English merger of Old English in (prep.) "in, into, upon, on, at, among; about, during;" and Old English inne (adv.) "within, inside," from Proto-Germanic *in (source also of Old Frisian, Dutch, German, Gothic in, Old Norse i), from PIE root *en "in." The simpler form took on both senses in Middle English.

Sense distinction between in and on is from later Middle English, and nuances in use of in and at still distinguish British and American English (in school/at school). Sometimes in Middle English shortened to i.

The noun sense of "influence, access (to power or authorities)," as in have an in with, is first recorded 1929 in American English. to be in for it "certain to meet with something unpleasant" is from 1690s. To be in with "on friendly terms with" is from 1670s. Ins and outs "intricacies, complications of an action or course" is from 1660s. In-and-out (n.) "copulation" is attested from 1610s.

Old English lætan (Northumbrian leta) "to allow; to leave behind, depart from; leave undone; bequeath," also "to rent, put to rent or hire" (class VII strong verb; past tense let, leort, past participle gelæten), from Proto-Germanic *letan (source also of Old Saxon latan, Old Frisian leta, Dutch laten, Old High German lazan, German lassen, Gothic letan "to leave, let"), from PIE *led-, extended form of root *‌‌lē- "to let go, slacken." If that derivation is correct, the etymological sense might be "let go through weariness, neglect."

The idea of slackening lies at the root of both applications of the term. When we speak of letting one go, letting him do something, we conceive him as previously restrained by a band, the loosening or slackening of which will permit the execution of the act in question. ... At other times the slackness is attributed to the agent himself, when let acquires the sense of being slack in action, delaying or omitting to do. [Hensleigh Wedgwood, "A Dictionary of English Etymology," 1859] 

He points to similar developments in French laisser "to let" from Latin laxare "to slacken," German lassen "to permit, to let," from dialectal lass "loose."

"The shortening of the root vowel ... has not been satisfactorily explained" [OED]. Of blood, from late Old English. Other Old and Middle English senses include "regard as, consider; behave toward; allow to escape; pretend;" to let (someone) know and to let fly (arrows, etc.) preserve the otherwise obsolete sense of "to cause to."

To let (someone) off "allow to go unpunished, excuse from service" is from 1814. To let on is from 1725 as "allow (something) to be known, betray one's knowledge of," 1822 as "pretend" (OED finds a similar use in the phrase never let it on him in a letter from 1637). To let out is late 12c. as "allow to depart" (transitive); intransitive use "be concluded," of schools, meetings, etc., is from 1888, considered by Century Dictionary (1895) to be "Rural, U.S." Of garments, etc., late 14c.

Let alone "abstain from interfering with" is in Old English; the phrase in the sense "not to mention, to say nothing of" is from 1812. To let (something) be "leave it alone" is from c. 1300; let it be "let it pass, leave it alone" is from early 14c. To let go is from c. 1300 as "allow to escape," 1520s as "cease to restrain," 1530s as "dismiss from one's thoughts." Let it go "let it pass, no matter" is as old as Chaucer's Wife of Bath: "But age allas Hath me biraft my beautee Lat it go, far wel, the deuel go ther with!" [c. 1395]. Let me see "show me" is from c. 1300.