I know that English has a deep orthography. I am wondering whether someone could tell me what the percentage of English words are governed by regular letter-sound rules? Thank you
This is a difficult question to answer, because it comes down to the rules you're using. At one extreme, you can give each letter a single pronunciation regardless of environment (the letters around it), which will have utterly abysmal accuracy. At the other extreme, you can have a separate rule for every single word in the OED, which will have 100% accuracy.
Clearly, neither of these extremes is particularly useful, and a better set of rules would lie somewhere in the middle. But it's hard to decide where in the middle you should draw the line. For example, "th" at the start of a word is pronounced /ð/ only in a handful of specific words, but some of those words are very common: this, these, that, those, etc. Is "hard-coding" these particular words being too specific? "Ch" before a vowel is /k/ only in foreign words; is it acceptable to have a rule that only applies in Greek-derived words, not Germanic ones?
For one data point, Mark Rosenfelder provides a set of rules for this (using a kind of awkward non-IPA transcription due to the limitations of Latin-1 encoding); he reports that his rules get it exactly right 59% of the time and pretty close 85% of the time on a test lexicon. How well native speakers actually follow these rules when pronouncing unfamiliar words is unknown, but potentially testable (though the execution of this is left as an exercise for the reader).
The direction of prediction is important. The letter-to-pronunciation mapping is much more regular than the pronunciation-to-letter mapping. The spelling of [sizr̩] is meaning-specific (Caesar, seizer, Seazer, Seezer) and since three of the extant spellings are personal names, it's not even a question of meaning, it's about which person you're referring to. But you know how to pronounce each of those spellings.
The degree of unpredictability is reduced if you use letter facts to generate etymological-source classes, so that <th,ph> won't be pronounces as [θ,f] in Zulu or Sanskrit words (if you can identify the etymology, perhaps from the spelling pattern). Still, there is an irreducible arbitrary core like the two pronunciations of Monticello (s, tʃ). Basically, the unpredictability depends on someone having actually worked out explicit rules: most linguists haven't written any such rules. If we had some set of words, we might try to compute the unpredictable residue, if we also had a dictionary that told us that Monticello can be pronounced two ways, Yakima can be pronounced two or three ways, and so on.