Identify the given numbers and the mean.

The given numbers are $1, 4, k, 4$ and $k+1$. There are $n=5$ numbers.
The mean is given as $k+1$.

The mean is the sum of all numbers divided by the count of numbers.

Substitute the given values:

Multiply both sides by 5:

Subtract $2k$ from both sides:

Subtract 5 from both sides:

Divide by 3:

The numbers are $1, 4, k, 4, k+1$.
Substitute $k = \frac{5}{3}$:

The mean is $k+1 = \frac{5}{3}+1 = \frac{8}{3}$.

The variance ($\sigma^2$) is the average of the squared differences from the mean.

Calculate the differences from the mean:

Sum of squared differences:

Now calculate the variance:

Simplify the fraction:

The standard deviation ($\sigma$) is the square root of the variance.

The value of $k$ is $\boxed{\frac{5}{3}}$ and the standard deviation is $\boxed{\sqrt{\frac{22}{15}} \approx 1.211}$.