I read the following:
For all $v\in H^{k+1}(K)$ we can associate a polynomial $p\in P_k$ (space of polynomial functions with degree $\leq k$), defined by $\forall \alpha \in N^n$, with $|\alpha|\leq k,\ \int_K \partial^{\alpha} p dx=-\int_K\partial^\alpha v dx$.
Why is this true? $v$ is not necessarily a polynomial here. The space $H^{k+1}(K)$ is a Sobolev space.
You are right: $v$ need not be a polynomial. But all we need from $v$ is the array of numbers $I(\alpha) := -\int_K \partial^\alpha v\,dx$, indexed by multiindex $\alpha$ with $|\alpha|\le k$. It does not really matter where these numbers came from: we can always find a polynomial $p$ of degree $\le k$ such that $\int_K \partial^{\alpha} p dx = I(\alpha)$ for all $\alpha$ with $|\alpha|\le k$.
The dimension of the space $P_k$ is equal to the number of multiindices $\alpha$ with $|\alpha|\le k$ (not surprising, considering that the polynomials are sums of monomials, which are in bijection with multiindices). Consider the linear map that sends every polynomial into the vector with components $\int_K \partial^{\alpha} p\, dx$. If $K$ has positive measure (as I suppose), this map has trivial kernel, since the highest nonzero derivatives that $p$ has are constants. By rank-nullity, this map is surjective.