Let $N=\binom{n+d}{d}-1$. Write $M_0,\dots,M_N$ for the monomials of degree $d$ in $x_0,\dots,x_n$. The Veronese embedding $$ v_d: \mathbb{P}^n\rightarrow \mathbb{P}^N $$ is the map sending $[x_0:\dots:x_n]$ to $[M_0(x_0,\dots,x_n),\dots, M_N(x_0,\dots,x_n)]$.
Let $v^*_d:k[y_0,\dots,y_N]\rightarrow k[x_0,\dots,x_n]$ be the map with $v_d^*(y_i)=M_i$ for every $i$. I have already proved that $P^n\cong v_d(\mathbb{P}^n)=V(\ker(v_d^*))$. Now I want to find the degree of $v_d(\mathbb{P}^n)$ (leading coefficient of its Hilbert polynomial).
To find the Hilbert polynomial I have to find $$ \dim_k (k[y_0,\dots, y_n]/\ker (v_d^*))_l $$ (the subscript means the homogeneous component of degree $l$).
Now $$ k[y_0,\dots, y_n]/\ker (v_d^*)\cong k[M_0,\dots,M_N]. $$
I would like to show that $$ k[M_0,\dots,M_N]_l=k[x_0,\dots,x_n]_{dl} $$ and hence that $$ \dim_k k[M_0,\dots,M_N]_l=\binom {n+dl}{n}. $$ A polynomial of degree $l$ in $k[M_0,\dots,M_N]$ is of course a polynomial of degree $dl$ in $k[x_0,\dots,x_n]$ and so this inclusion is easy.
I can't show the other one. Maybe I can also prove that the set of all monomials of degree $l$ in $K[M_0,\dots,M_N] $ is exactly the set of all monomials of degree $dl$ in $k[x_0,\dots,x_n]$. But I have the same problem. The degree should be $d^n$.
Let's get this off the unanswered list. As Mohan suggests in the comments, induction on $l$ can solve this problem.
Base case, $l=0$: the degree-zero polynomials in $k[M_i]$ are the constants, which are also the degree-zero polynomials in $k[x_i]$.
Inductive step: suppose we know the claim for $l=l_0$ and we want to prove it for $l=l_0+1$. Write a monomial of degree $d(l_0+1)$ in $k[x_i]$ in lexicographical form, i.e. $x_0^{a_0}x_1^{a_1}\cdots x_n^{a_n}$. Let $j$ be the smallest integer so that $\rho=\sum_{i=0}^j a_i\geq d$. Then $x_0^{a_0}\cdots x_j^{a_j-\rho+d}$ is of degree $d$ and divides $x_0^{a_0}x_1^{a_1}\cdots x_n^{a_n}$, so we can write $$x_0^{a_0}x_1^{a_1}\cdots x_n^{a_n}=(x_0^{a_0}\cdots x_j^{a_j-\rho+d})\cdot p$$ where $p$ is of degree $dl_0$. As $x_0^{a_0}\cdots x_j^{a_j-\rho+d}$ is a monomial of degree $d$, it is one of the $M_i$, and by the inductive hypothesis, $p$ is a product of $M_i$. So $x_0^{a_0}x_1^{a_1}\cdots x_n^{a_n}$ is a product of $M_i$, and we've proven the claim.
As you note in your problem, this shows that the dimension of the $l^{th}$ graded piece of $k[M_i]$ is $\binom{n+dl}{n}$. Expanding this binomial coefficient, we find that it is of the form $$\frac{(dl+n)(dl+n-1)\cdots(dl+1)}{n!}$$ which has leading term $\frac{1}{n!}d^nl^n$ and therefore the Veronese embedding has degree $d^n$ as desired.