are you assuming that ? i'm saying this because does not imply that , even in prime rings. if you're assuming , why do you need the condition ? are you trying to prove that a prime ring has no non-zero nilpotent ideal? to prove this, you only need to be semiprime.
anyway, is obviously a right ideal. i don't know why you can't prove this! it is non-zero becasue and so i guess you forgot to mention in your question that and .
for the second part of your question, you don't need Herstein! suppose that is a right ideal of a ring and for some integer . then is clearly an ideal of and
thus and so is a nilpotent ideal of . note that if , then because