That whole thing is a mess.
For one thing, as you said, it's not 100 percent clear Elvis is not "the nicest person I know."
Furthermore, if instead Kevin is "the nicest person I know, the "modifier touch rule" is that the modifier has to touch
a noun or noun phrase that it modifies, not just a noun. So, since "my friend Kevin, who likes Elvis" is noun phrase, the modifier "the nicest person I know" could indeed be touching what it modifies, which in that case would be "my friend Kevin, who likes Elvis." In that case, the sentence may not be ideal because of the way the second modifier, "the nicest person I know," follows "who likes Elvis," but the sentence is arguably OK.
This rule is covered in the Modifiers chapter of the
Target Test Prep Verbal Course, which is super comprehensive and includes over 70 practice questions on Modifiers alone.
You can check out that chapter and the rest of the
Target Test Prep course by signing up for the $1 trial.