Um, yeah.
I don't think it's particularly meant to be a secret, either. Well, maybe within the context of the Durarara!! universe, where Izaya wants to pretend he is in complete control of everything he does and feels, but from an outsider point of view? I mean, duh. I'm a psychology major, so I kind of had to make a page like this on here, or else I would feel unsettled.
Getting to the question of what mental illnesses he has, exactly, is a bit trickier. 'Illnesses', because there's no way he only has one. For now, I'll start with what I think would be the first to develop:
C-PTSD (aka Complex PTSD): Now, I'm certainly not an expert in any of these. I just happen to know about them because I like to read the DSM for fun (among other things...). So don't take anything I say here as concrete medical facts. I'm just relating it to my anime wife.
Anyways, C-PTSD is quite different from typical PTSD. Instead of forming from one singular traumatic event, C-PTSD forms from trauma that occurs over a much longer timeframe... you know, like your entire childhood (the most crucial years of development). In Izaya's case, his parents often left him alone at home while leaving to go abroad for work. While they may have been "good" parents while they were around, the instability and unreliability of both of them would have likely left a strong impression on Izaya.
Adding to this, when Izaya was only 9, they decided for some reason that they would do the entire thing again, despite their busy schedules (can you tell I really hate them). So then here is this child stuck caring for not only himself, but now two infants. The stress from both this and having no real parental figures to safely rely on would have been more than enough to force him to grow up faster than he needed to.
This is a different train of thought, but I can't help but consider the fact that Izaya was essentially recruited by the Yakuza at such a young age to play a factor in this as well. Maybe not to the extent that the above mentioned stuff did, but I don't think it's wrong to say he was groomed into it. He had no business working with such a shady group, and all the adults that were involved in that process should feel ashamed.
Borderline Personality Disorder (aka BPD) (and the entire Cluster B of PDs in general): Now here's where I might lose a few people. That is fine. If you are not smart enough to understand it, this may be a good starting point for you.
In my opinion, the main thing with BPD is instability. That goes for both the self and for the individual's life. I think Izaya absolutely fulfils this criteria. Hell, he's been described as unstable in the novel.
Izaya is also suicidal. Will he ever kill himself? Absolutely not, because he is much too cowardly to do it himself. He even admits this himself. So by having Shizuo kill him, he not only accomplishes his goal of dying, but he also proves that Shizuo truly is a monster now that he has blood on his hands.