Is the grammar of this sentence correct?

English language is loaded with non-standard phrases that are rooted so deeply in its vocabulary that at times they cannot be distinguished from the accepted Standard English terms.

Since "rooted so deeply" qualifies "phrases", you should use "that" instead of "which".

Might I suggest :
The english language is loaded with non-standard phrases, some of which are rooted so deeply in its historical and current usage that at times they cannot be distinguished from the standard phrases of english.

Actually, they can. It is difficult to reword a sentence without its contextual meaning.

That is an errant prescription which doesn't describe how English works, Engineer. Both that and which can be used

those that are rooted so deeply is indistinguishable from those which are rooted so deeply so those which chose that those which are rooted so deeply over those which chose that that which are rooted so deeply are those that chose that those which chose that that which are rooted so deeply are deeply rooted and thats that

Something hallucinogenic must have gotten into your oolong tea, Oolong.