I wouldn't necessarily put it that way, but that sentence sounds fine grammatically - 'good English' is what the speaker speaks (and therefore the sentence is self-referential…).
Could this also been translated by 'I speak good English ' ? It's not a word for word translation but to me it sounds more natural .
Good is an adjective, while well is an adverb. In this case, you're modifying "speaking" telling us how you speak. Since speaking is a verb, you would use "well" or хорошо.
I speak English good, no excellent, and with new grammar.
Я говорю по-английски хороший, нет, даже отличный, и с новой грамматикой
[ The preceding are deliberate non standard English translated into non standard Russian ]
Adjective ‧ en.oxforddictionaries.com/definition/good
Good is an adjective. It modifies (describes) a noun. In each case, the adjective good is modifying a noun. Good should never be used as an adverb (modifying a verb). ‧ webapps.towson.edu/ows/good__well.htm
In both casual speech and formal writing, we frequently have to choose between the adjective good and the adverb well ‧ guidetogrammar.org/grammar/adjectives.htm
[ The following "" are non standard English ]
Speaking not too "good" is not good speech.
Speech not too "well" is also not good speech.
While in colloquial English it's common, it's a big no-no in formal English.
I'm gonna have to agree. JWolf88's explanation as well as Hojikie's observation is spot on, and should therefore not be included as an acceptable answer.
And how about "I can speak English well"? Is there a difference in meaning or why isn't it accepted?
Would that not be, "Я очень хорошо говорю по-английски" or a similar phrase?
2 letters - английски_. One letter you missed is the й, and you have one too many и's at the end. Also, I am not sure, but you may need to hyphenate. I have not tried to not hyphenate, but I assume that it is necessary, considering that this is how these words are spelled in the dictionary.