Ah -- the Ukrainian pains of asylum.
A couple of pointers about applying for asylum when you are already in the US. First, affirmative asylum (that's the kind you apply for when you aren't in the hot seat, i.e. in front of the immigration court!) must be done in the first year you are in the country. Sadly, the government does not want nor approve opportunistic "squatting" in the US -- hanging out here in the States until something bad happens at home, and then...poof! Asylum time! So, as for affirmative asylum, you have a great shot if you're in the first year of your presence a la United States. Otherwise...nada. See 8 U.S. Code § 1158 for information on asylum and the process of applying.
