"Who" and "that" would be fine.
Not sure about "whom." The fact is that it can only be used as an object:
○ This is the girl whom I will visit.
× This is the man whom will take care of you. (○ who will take care of you)
The question is then whether a copula can take the accusative case. After some research, it turns out that even though this often happens in practice ("He used to be him"), strictly speaking only the nominative is allowed ("He used to be he"). As such, "whom" should probably be avoided in this case. There's another reason too: "whom" sounds a little stilted and its use is on the decline in favor of "who", as JuliMaruchan pointed out.
"Which" is used for things, not people, and is therefore most definitely not valid here.
That said, the most natural-sounding option is to use the zero relative pronoun - that is, to put nothing in the blank at all (as johnnyG already suggested).