Besides indict and its compounds, the only other word I know of with 〈ict〉 pronounced /aɪt/ is deictic, which is pronounced /ˈdaɪktɪk/. (It has various compounds, too: anapodeictic, endeictic, epideictic.) But that’s because of the 〈ei〉 not the 〈ict〉.

What’s going on with indict is different. It used to be endite or indite. ɴᴏᴛᴇ ᴡᴇʟʟ: We got it from the French without any c in it in the first place. So we never said what wasn’t there. (And in fact indite still exists but now with a somewhat different meaning.)

Regarding indict, though, the OED2 writes that there seems to have been some confusion along the way:

indict /ɪnˈdaɪt/, v.¹

Forms

• ɑ. 4–6 endyte, 4–7 endite, 6 endight (endict).
• β. 4–9 indite, 5 indyte, (6 indyght, 6–7 indight).
• ɣ. 7– indict.

Etymology

ME. endite-n, a. AFr. endite-r to indict, charge, accuse, corresponds in form to OFr. enditer, -ditier, -ditter, answering to a late L. type *indictāre, f. in- (in-²) + dictāre to say, declare, dictate. But the OFr. verb is recorded only in the senses ‘make known, indicate, dictate, suggest, compose, write, instruct, inform, prompt, incite’ (Godef.), so that the history of the AFr. and ME. word is not clear. A corresponding med.L. indictāre to indict, accuse, is cited by Du Cange only in English legal use, and seems to be merely the latinized form of the AFr. and ME. verb, in accordance with which again the ME. endite has been altered to indite, and (since 1600) written indict, though the spoken word remains indite. See also indite v.

The sense of endite, indict, may have arisen from L. indīcĕre ‘to declare publicly’, taken as in Ital. indicere ‘to denounce’ (Florio); but it comes near to a sense of L. indicāre to indicate ‘to give evidence against’; and it appears as if there had been, in late L. or Romanic, some confusion of the L. verbs indicāre, indīcĕre, indictāre: thus in Ital., Florio has ‘Indicare, to shew, to declare, to utter; also to endite and accuse, as Indicere’; ‘Indícere, to intimate, denounce, manifest, declare;··also to accuse, to appeach or detect’; ‘Indittare, to indite; also as Indicere’; ‘Indittore, an inditer, a denouncer; also an intimator’.

So along with various words like debt and island, indict got dubiously “Latinized” by putting in a still/now-silent letter. The pronunciation never changed.