Damicci said:
なんか
Can be used like なんとなく？
たとえば：
あ！なんか、この問題を分かるのが出来ます。
（Ah! Somehow I am able to understand this problem.)
なんか彼はまだその仕事をつづけます。
(Somehow he is still working that job.)
Ah, you have come across something good here! They are definately very similar, and unfortunately, all I can say is that they seem to have a slightly different nuance, although I can not really explain that difference. I'll try, though. Also keep in mind that what follows is nothing but my own interpretation of the language, and I may not necessarily be correct.
In your first example, which I might rephrase as
なんか、この問題わかった！
I think it is gramatically correct to use nannka here. But the nuance is different. nannka would seem to me to express almost surprise that you figured it out, but nanntonaku would seem to not have that feeling, but rather just relate that you seemed to just somehow be able to figure it out.
In the second example, it seems to me that if you use nannka, you would have to add youda at the end, like,
なんか、彼はまだその仕事を続けているようだよ。
Don't ask me why I get this feeling, but I do. Although I must admit that by doing this, the connection of "somehow" and him continuing the work/job seems to be weaker, or nonexistant. Instead it would be like, "I don't know. He seems to still be at that same job." At least this is how I feel. If you were to put nanntonaku instead, the connection to him continuing is much stronger. Here I get the impression that he's kind of just "sliding by" somehow at his job.
ような？
えっと たとえば：
A: おい！ あの人はような顔がありませんか？
(Hey doesn't that person seem familiar?)
B: だれか？ あ！そう思いますよ。
（Who? Ah! Yes I think so too!)
I might rephrase this as
A: あの顔はなんか見たような気がするけど。
B: うん。そうだね。
Your sentence, if you change the あの人はような to あの人のような, sounds more like to me, "Do you have any faces like that guy's?" Something you might say at a store, but not.