The lemma comes down to proving $P(\|\frac{A}{\sqrt{k}}v\|^2>(1+\epsilon)^2)\in O(\frac{1}{n^3})$, $A_{ij}\sim N(0,1)$, A is $k$ by $n$ matrix, $\|v\|=1$. You may want to follow the proof of http://www.wisdom.weizmann.ac.il/~robi/teaching/2009b-SeminarGeometryAlgs/scribe4.pdf
Why Markov's inequality does not give a tight upper bound to prove Johnson-Lindenstrauss Lemma? For example, for some $N$
$$P(\|\frac{A}{\sqrt{k}}v\|^N>k^{\frac{N}{2}}(1+\epsilon)^N)\le\frac{E\|\frac{A}{\sqrt{k}}v\|^N}{k^{\frac{N}{2} }(1+\epsilon)^N}$$