Some version of Hahn-Banach theorem 
I understand the first part of the theorem. I can prove the part about $\delta$ if the inf is actually achieved. But I wonder whether this is even true.

Consider $Z=Y\cup\operatorname{span}\{x\}$. This is obviously a subspace. Now define $f:Z\to \mathbb K$ by
$$f(z)=\lambda\delta,$$
where $z=y+\lambda x$ ,remembering the definition of $Z$. I leave it to you to show that $f$ is bounded and linear, $\|f\|=1$, and $f(x)=\delta$, but feel free to ask hints if you get stuck. We can then use the standard Hahn Banach theorem for normed spaces to extend $f$ to a map $\tilde f\in X^*$ satisfying the requirements.
To show that $\|f\|=1$, we first consider some $z=y+\lambda x$ with $\lambda\neq 0$. Then
$$|f(z)|=|\lambda|\inf_{y'\in Y}\|x-y'\| \leq |\lambda|\|x-(-\lambda^{-1}y)=\|z\|.$$
Hence $\|f\|\leq 1$. For the reverse inequality, remember by definition of the infimum there exists a sequence $(y_n)\subset Y$ such that $\|y_n-x\|\to \delta$. Set $z_n=y_n-x$. Then $f(z_n)=-\delta$ for all $n\in\mathbb N$. Use these facts along with the definition of the norm of a functional to derive the desired inequality.
