I suppose one question is why suppose that there is an "I" there to know. Lots of philosophers have rejected this very Cartesian assumption. Hume, Sarte, Dennet all rejected the idea. Not a criticism; just a question. Here is suggestion. Start with simpler problems first. You want to know what it is to know yourself, but what is it to know anything at all? What is it to know your dog? Your house? I guess my point is that sometimes we ask "what is it to know myself" and we don't even know what we mean by "know".
@urizenus, I understand that there are a lot of things to be known and that what we actually mean by "known" is also unknown. For example, knowing something means to get till the roots and once we get to that point, there has to be the end of science and then that's where philosophy comes into the picture. For example, knowing about dogs doesn't actually mean to 'know' the breed or name of that dog but to understand the importance of their existence or the origin of their existence.
Definitely, knowing things should start from simpler things. And to know what it means to know anything at all. :)