How do I use the pumping lemma for a^n b^m a^(n+m) ? How can I choose the pumping length?
It seems that you are trying to force the string length to be $p$, which is not necessary. What is required is that $|xy| \leq p$. The length of $w$ can be greater than $p$. Also, $n$ or $m$ can be $p$ or some multiple of $p$, whatever  makes the reasoning for the impossibilty of partition of $w$ to satisfy the lemma. With that, I think you can follow through and complete your proof.
Just remember, the aim is not to look for a partition of $w$ that does not work. Rather show that no partition works.
Cheers.