Stinespring's dilation theorem is well known for unital $C^*$-algebras: not only it allows us to recognize any completely positive map $\varphi:A\to B(H)$ as the dilation of a non-degenerate representation $(K,\pi)$ of $A$, (i.e. we have $v\in B(H,K)$ s.t. $\varphi(a)=v^*\pi(a)v$ for all $a\in A$) but the dilation can be minimal, that is the triple $(K,\pi,v)$ can be taken to satisfy $K=[\pi(A)vH]$. I demonstrate how this can be done since this is what we care about here:
Let $(K,\pi,v)$ be a dilation of the c.p. map $A\xrightarrow{\varphi}B(H)$. Set $K_1=[\pi(A)vH]$. This space is invariant for the action of $\pi(A)$, so we may set $\pi_1(a):=\pi(a)\vert_{K_1}:K_1\to K_1$ and it is routine to satisfy that $\pi_1:A\to B(K_1)$ is a unital $*$-homomorphism (it is unital because $\pi$ is also unital). Here is the important part: since $\pi(1_A)=id_K$, we have that $vH\subset K_1$, so $v$ acts as $v:H\to K_1$ and we still have $\varphi(a)=v^*\pi_1(a)v$ for all $a\in A$, so $(K_1,\pi_1,v)$ is a dilation of $\varphi$ satisfying $K_1=[\pi_1(A)vH]$.
Now there is a difficult proposition (i.e. Brown and Ozawa, C*-algebras and finite dimensional approximations, prop. 2.2.1) that says the following:
Let $A$ be a non-unital $C^*$-algebra, $B$ a unital $C^*$-algebra and $\varphi:A\to B$ a contractive completely positive map. Then $\varphi$ extends to a unital c.p. map to the unitization $\tilde{\varphi}:\tilde{A}\to B$ given by $\tilde{\varphi}(a\oplus\lambda)=\varphi(a)+\lambda1_B$.
Now this can be combined with Stinespring's theorem and it yields a Stinespring-like theorem for non-unital $C^*$-algebras:
If $A$ is non-unital and $\varphi:A\to B(H)$ is a c.p. map, then we may find a representation $(K,\pi)$ of $A$ and $v\in B(H,K)$ such that $\varphi(a)=v^*\pi(a)v$ for all $a\in A$.
The problem is I cannot obtain a minimal dilation here. If I set $K_1=[\pi(A)vH]$, then I cannot get $vH\subset K_1$. The representation $(K_1,\pi_1)$ of $A$ is non-degenerate, so if $(u_\lambda)$ is an approximate unit of $A$, then $\pi_1(u_\lambda)\to id_{K_1}$ in SOT. This does not help though, because we do not know a priori that $vH\subset K_1$. On the other hand, the representation $(K,\pi)$ is simply the restriction of $(K,\pi')$ which is a non-degenerate representation of $\tilde{A}$. Of course this does not mean that $(K,\pi)$ is non-degenerate for $A$. Do you know how I can get through this?
Since $\pi(A)$ is a C$^*$-algebra, $\pi(A)''$ is a von Neumann algebra. Because $\pi$ is non-degenerate, by the Double Commutant Theorem there exists a net $\{a_j\}\subset A$ such that $\pi(a_j)\to I$ sot. Then for any $\xi\in H$ you have $$ v\xi=\lim_j\pi(a_j)v\xi\in K_1. $$