Well, your hypothesis is that
$$
A=k[y_1,\dots,y_p] \text{ is local, }\mathfrak m^N=0, k=A/\mathfrak m \text{ and }
\mathfrak m/\mathfrak m^2= k<\overline{x}_1,\dots,\overline{x}_n>
$$
By Nakayama $\mathfrak m=(x_1,\dots,x_n)$. The chain
$$
0 \subseteq \mathfrak m^N \subseteq \mathfrak m^{N-1} \subseteq
\mathfrak m^2 \subseteq \mathfrak m \subseteq A
$$
then gives you the finite-dimensional $k$-vectorspaces $\mathfrak m^i/\mathfrak m^{i+1}$, so by induction it follows that $A$ itself is a finite-dimensional $k$-vectorspace [EDIT:] generated by $1$ and a finite set of monomials of the $x_i$