Ned Stark finding out the bastard - mixed scenes?

Different bastards, different context

There are multiple bastards in the story, and the two scenes you highlight focus on completely different bastards.

He's found one bastard already.

This is in reference to Robert's bastards, of which Gendry is one but Robert had many others (who were later all killed except Gendry, I believe?). I think he was talking about Gendry specifically; but even if I'm wrong about that, he was definitely speaking about a bastard that Robert fathered with another woman.

More elaboration on the Varys/Illyrio conversation can be found here.

The point here is that the referenced bastard (Gendry or others) is a child of Robert Baratheon.

"Joffrey Baratheon.. golden hair."

This proves to Ned that Joffrey is not genetically a Baratheon, and therefore a bastard since his mother had him while married to Robert, who is not the father. The only logical answer is that someone else is his father, and thus he is a bastard.

The point here is that the referenced bastard (Joffrey) is NOT a child of Robert Baratheon.

This is at least what happens in the books, so it passes for in-universe canon. I don't know how much can be inferred from the show and if it's consistent, but here goes:

1. Ned inquired and found out Jon Arryn has asked for 'the book' shortly before death. Such an inquiry of an otherwise uninteresting book piqued his curiosity and he borrowed the book.
2. Ned meets Gendry, who resembles Robert Baratheon very much (including the hair). He doesn't know for sure he's Robert's bastard (Gendry doesn't either, as far as I remember)
3. They discuss how Ned is gathering clues and getting closer to an epiphany.
4. Ned has the epiphany, but it's too late for Robert by then.

Ned's epiphany is that the darker Baratheon hair is dominant. Robert had some sixteen bastards already by then, but Ned hadn't met them all. There are more puppetmasters following the way the events unfold, and they're following the action very closely.