2nd application-need to see ill gran

[kingjamesuk]

THIS IS MY FIRST THREAD.

I recently was refused a visa under 214(b).
I have to apply due to the fact that i overstayed my waiver visa, 5 years ago. It was by 2 months.
Anyway, i was refused a visa 3 weeks ago, due to the fact that i had a lack of evidence for the purpose of my visit. I intend to see my uncle and Grandmother in California, who have both had strokes, and are becomming progressively more ill.
Now i have all the information from my relatives that has been requested by the embassy in London. This includes doctors letters etc.
Is the fact that i am visiting my ill relatives, enough to be issued a visa? Or, is the embassy just playing with me, and has no intention of giving a visa at all? Does over staying by 2 months automatically make me inadmissible for the rest of my life?
My relatives are furious about it, as this will be the last chance i ever get to see my grandmother, because she wont be around forever. I just got the feeling that the american government just wants to stop everyone they possibly can now, regardless of status or genuine intentions. My next appointment is in a month time. Also, do they issue a standard 6 month visa, or is it time restricted to the length of the intended stay in america?
Many thanks for any helpfull replies!