It is spelled that way because of an error that stuck around.
What dictionary did you consult that failed to explain this? Please tell us, so that we can steer people clear of it in the future.
The OED says of delight the noun:
ME. delit, a. OFr. delit (-eit), (= Pr. deliet, Sp. deleite, Ital. diletto), f. stem of deliter vb.
The etymological delite is found as late as 1590, but earlier in 16th c. it had generally been supplanted by delight, an erroneous spelling after light, flight, etc.
And it says this of delight the verb:
ME. delite-n, a. OFr. delitier (-leitier, -leter, -liter) = Pr., Sp. delectar, Sp., Pg. deleitar, Ital. delettare, dilettare:-L. dēlectāre to allure, attract, delight, charm, please, freq. of dēlicĕre to entice away, allure: cf. delicious.
The current erroneous spelling after light, etc. arose in the 16th c., and prevailed about 1575: the Bible of 1611 occasionally retained delite.
The oldest noun uses of delight are:
- A. 1225 Ancr. R. 272 ― So sone so me··let þene lust gon inward & delit waxen.
- A. 1225 Ancr. R. 102 ― Þes cos··is a swetnesse & a delit of heorte.
- C. 1230 Hali Meid. 7 ― And habbeð mare delit þerin þen anie oðre habbeð i likinge of þe worlde.
- A. 1240 Ureisun in Cott. Hom. 201 ― Þe muchele delit of þine swetnesse.
- A. 1300 Cursor M. 23339 (Cott.) ― Bot suld þai haf a gret delite, To se þam setlid in þair site.
- C. 1340 Cursor M. 8164 (Fairf.) ― Þai hailsed him wiþ grete delite.
- C. 1385 Chaucer L.G.W. 1199 Dido, ― With sadyl red enbroudit with delyt.
- C. 1386 Chaucer Prol. 335 ― To lyuen in delit was euere his wone, For he was Epicurus owene sone.
While the oldest verb uses are:
- A. 1225 Ancr. R. 52 ― Eue··iseih hine ueir, & ueng to deliten i þe biholdunge.
- C. 1300 K. Alis. 5802 ― So hy ben delited in that art That wery ne ben hy neuere cert.
- 1303 R. Brunne Handl. Synne 3086 ― Ȝyf þou delyte þe oftyn stoundes, Yn horsys, haukys, or yn houndes.
- A. 1325 Prose Psalter l[i]. 17 ― Þou ne shalt nouȝt deliten in sacrifices.
- C. 1340 Cursor M. 1560 (Fairf.) ― A-mong caymys kyn, þat delitet ham al to syn.
- C. 1374 Chaucer Anel. & Arc. 266 ― But for I··was so besy you to delyte.
- C. 1385 Chaucer L.G.W. 415 ― Yet hath he made lewde folke delyte To serue yow.
Even Chaucer back in the 14th century varied his spelling of the word, alternating between using a y or an i, and whether to use a final vowel. By the time the 16th century rolled around, the word we now spell as light had many spellings, including in particular lyte. It is quite likely that delight was caught up in that same spelling change.
As for the pronunciation, I am not sure what you would be expecting there. It is not a new import from Latin like infect is, having gone through Old French and lost its -c-. I can’t think of many English words that sound like that but which came from a Latin -ect-, but with despite and indict being two possible examples. Things like incite and recite never had the -ct-.
Parallel, perhaps, to delight is the legal term, delict, from Latin dēlictum. It is pronounced /dɪˈlɪkt/, and originated in the 16th century. It is perhaps most commonly seen in the phrase in flagrant delict, which is translated either from the Latin in flagrante delicto or from the French, en flagrant délit.