Remember that 'Not only ..." is always followed by an inversion construction when it begins a sentence e.g.
Not only was I wet and lonely, but I was lost, too.
Not only did she bring Paul, but she brought all her family, too.

Since the second half of the sentence is negative with 'not. . . either', the verb in the first half will almost certainly be negative, too.

The subject in the second half is 'he', so the subject in the first half will almost certainly be 'he'.

The tense in the second half is the Simple Past, so the tense in the first half will almost certainly be the Simple Past.

In the second half, 'her second name' gives a clue or pointer to 'her first name' needed in the first half.

Such reasoning above therefore suggests that the first half of the sentence should read:
Not only did he not know her first name, ...

Or perhaps, by the same kind of reasoning:
Not only could he not remember her first name, ...

The have and the 've ... already' tell you that the tense needed is Present Perfect.

The verb 'have' in the question suggests that the main verb needed in the answer is also 'have' i.e. 'have had',  not 'have drunk/consumed .

The phrase 'another cup of tea' tells you that the second speaker must have had at least one already, and possibly more. Therefore the speaker has probably had 'enough' or 'plenty' or 'more than enough', since he says 'No, thank you'.

Such reasoning above suggests that the sentence should read:
I've had enough already.
I've had more than enough already.

Or perhaps, by the same kind of reasoning:
I've had plenty already.
I've had one already.