Handling dead state in NFA to DFA conversion
Your DFA actually looks correct. Why do you need a dead state?

A dead state means the automaton was given a prefix of an input that will never lead to an accepting state. But the language of the NFA has no such prefixes - whatever the prefix is, if you add "10" or "11" you will get  to an accepting state.

Anyways, the right thing to do now is to verify that the DFA works fine for several sample inputs. More generally, to prove it actually does decide the language generated by $(0+1)^*1(0+1)$.