Computing the exponential generating function of the Bell numbers.

There's an error in your second line. Using $B_0=1$, you should get $$B(x)=1+\sum_{n=0}^\infty\sum_{k=0}^n\binom nkB_k\frac{x^{n+1}}{(n+1)!}.$$ I would now differentiate....

$$\sum_{n=k}^\infty \frac{x^n}{k!(n-k)!} = \sum_{n=0}^\infty \frac{x^{n+k}}{k!n!} = \frac{x^k}{k!} \sum_{n=0}^\infty \frac{x^{n}}{n!} = \frac{x^ke^x}{k!}$$