I've counted one in the secondary hull. I've counted zero, one and maybe two more in the saucer.

So far, we've guessed one or three M/AM reactors. We're just as vague as the originally writers.
I think there are at least two engine rooms per Scotty’s line in “The naked Time”. But to really shed some light on the problem we need to disentangle some terms that many are wont to conflate i.e.;
Engineering
Deck is not the same as Engineering
Section, which in turn is not the same as Engineering
Hull.
The term “Engineering Hull”
is not a TOS canon term, and “Engineering Section” is not a substitute for it. “Mirror, Mirror”, “Journey To Babel”, and “I Mudd” all indicate in various ways that the Engineering
Section is a specific point
within the ship and so in no way should it be construed as a large external area such as the Secondary Hull. The closest we come to any canon references to that area of the ship is the occasional references to “lower decks” or “lower levels”.
Among the “lower levels” is to be found the Engineering
Deck as per “The Enemy Within”; as well as the Hangar Deck as per “The Omega Glory”, so it’s reasonable to assume that the” lower levels” is a general term referring to those deck/levels in the Secondary Hull, and that therefore the Engineering
Deck is in the Secondary Hull.
So far so good, however;
But it is also made abundantly clear in “The Omega Glory” that Kirk is still in the Exeter’s Engineering
Section when Spock reports via communicator that he and another crewman are on the “lower levels” thus distinguishing his location within the ship from that of Kirk’s. Furthermore, this episode is the one which features a clip of “an” engineering room, inserted among other stock footage, to represent empty areas of the ship during Kirk’s intercom call to any survivors aboard, and during which, significantly, Kirk reiterates that he is in the Engineering
Section. So it is not unreasonable to conclude that there are at least two engine rooms and that one is in the Engineering
Section which
is not among the lower levels (and thus not in the Secondary Hull), while the other engine room, which is AKA the “Engineering Deck”
is located down there.
So if the Engineering
Deck is in the lower levels, but the Engineering
Section is not in the Secondary Hull then where is the Engineering Section?
In “The Ultimate Computer” we are told that one of the other ships was hit in its
Engineering Section, with possible damage to her
impulse engines. Therefore it seems clear that the Engineering Section is in the impulse deck, just where TMOST and FJ put it. I know this is an uncomfortable idea for many fans but it fits the factoids we know and requires the least amount of assumptions and hand waving.