We do in a way; it's just that our letters with bits added above and below them have come to be considered separate letters.
The same happened in Swedish, where "ä", "ö" and "å" are considered separate letters of the alphabet rather than letters with accents or diacritics. Similarly Danish with "æ", "ø" and "å".
In English (and other languages), "j" was originally an "i" with a tail beneath it. The OED tells us:
From the 11th to the 17th cent., then, the letter I i represented at once the vowel sound of i, and a consonant sound /dʒ/, far removed from the vowel. Meanwhile, the minuteness and inconspicuousness of the small ı, and its liability, especially in cursive writing, to be confounded with one of the strokes of an adjacent letter, had led in mediæval Latin and general European writing, and thus also in English, to various scribal expedients in order to keep it distinct. (See I.) Among these, an initial ı was often prolonged above or below the line, or both; a final ı was generally prolonged below the line, and in both cases the prolonged part or ‘tail’ came at length in cursive writing to be terminated with a curve... The ‘dot’, used to individualize the minuscule i, was also used with the tailed form, and thus came the modern j, j.
Historically, in Old English, ash (æ), thorn (þ) and eth (ð) were added to the alphabet, but they dropped out sometime after the Norman invasion and the subsequent shake-up of English orthography. (While æ is occasionally still seen, it is now just another way of writing "ae", and it isn't very common any more even in that use.) Old English scribes also placed length marks over long marks on at least some occasions (see here).
As others have noted, English has a number of words with accents ("café", "résumé"), but of course, these accents are often considered optional, and most of the words they appear in still feel "foreign" as well.