This is a fleshed-out version of Chris Eagle's answer/comment, except that we could avoid the machinery of Dirac's theorem. I recall a result from a related question of mine:
Theorem A. If $|V(G)|$ is even and $\delta(G) \geqslant \frac{|V(G)|}{2}$, then $G$ contains a perfect matching.
In fact, we will not need a proof of the above theorem, just its statement. [That said, I encourage you to read joriki's elegant direct proof that just uses Hall's theorem.]
The present question is a generalization of this theorem. Define a $b$-factor of $G$ to be any spanning subgraph $H$ of $G$ that is $b$-regular. (Plug in $b = 5$ to get the desired statement.)
Theorem B. If $|V(G)|$ is even and $\delta(G) \geqslant \frac{|V(G)|}{2} + b -1$, then $G$ contains $b$-factor.
To prove this, we repeatedly apply "Theorem A" $b$ times, each time deleting the resulting perfect matching from the graph. In each iteration, each degree goes down by $1$, so the condition that $\delta \geqslant \frac{|V(G)|}{2}$ is preserved at each of the $b$ steps.