In these formulas, we hav that $X = (x(t), y(t), z(t))$. Therefore, we begin by calculating the derivatives of the vectors component-wise with respect to $t$ and have
$$X' = (x'(t), y'(t),z'(t)) = \left( e^t \sin(t) + e^t \cos(t) , e^t \cos(t) - e^t \sin(t), e^t\right)$$
$$X^{\prime\prime} = \left(2e^t \cos(t) , - 2e^t \sin(t), e^t\right)$$
$$X^{\prime\prime\prime} = \left( 2e^t \cos(t) - 2e^t \sin(t), -2e^t \sin(t) - 2e^t \cos(t) , e^t\right)$$
To compute the curvature, we start by computing
\begin{align} \left| X' \times X'' \right|
& = \left| (e^{2 t}\cos(t)+e^{2 t} \sin(t), e^{2 t} \cos(t)-e^{2 t} \sin(t), -2 e^{2 t} \cos^2(t)-2 e^{2 t} \sin^2(t)) \right| \\
&= \sqrt{6}e^{2t}\end{align}
Also, $$v^3 = \left| X' \right|^3 = \sqrt{(e^t \sin(t) + e^t \cos(t))^2 + (e^t \cos(t) - e^t \sin(t))^2 + (e^t)^2}^3 = 3\sqrt{3}e^{3t}$$
Therefore we have that the curvature is given by $$\kappa = \frac{\left| X' \times X'' \right| }{v^3 } = \frac{\sqrt{6}e^{2t}}{3\sqrt{3}e^{3t}} = \frac{\sqrt{2}}{3}e^{-t}$$
For the torsion, we have a few more calculations. Now
$$ (X' \times X^{\prime\prime} ) \cdot X^{\prime\prime\prime} = (e^{2 t}\cos(t)+e^{2 t} \sin(t), e^{2 t} \cos(t)-e^{2 t} \sin(t), -2 e^{2 t} \cos^2(t)-2 e^{2 t} \sin^2(t)) \cdot ( 2e^t \cos(t) - 2e^t \sin(t), -2e^t \sin(t) - 2e^t \cos(t) , e^t ) = -2e^{3t}$$
Also
$$v^3 \kappa^2 = 3\sqrt{3}e^{3t} \left( \frac{\sqrt{2}}{3}e^{-t} \right)^2 = \frac{2\sqrt{3}}{3} e^t$$
Therefore the torsion is $$\tau = \frac{-2e^{3t}}{\frac{2\sqrt{3}}{3} e^t} = -\sqrt{3}e^{2t}$$
The radius of curvature is larger than the radius of the cylinder because the helix is climbing as it winds. That means the best matching circle will be larger than the circle of the cylinder.
If you imagine a helix that climbs very steeply you can see that it's locally nearly a straight line, so tangent to a very large circle.
Best Answer
Curvature and torsion are independent of the location of the curve so we can ignore those factors. For a circular helix of radius $r$ and pitch $2\pi p$, we can parameterize it as follows:
$x(t) = r\cos(t),\, y(t) = r\sin(t),\, z(t) = pt.\,$
The curvature for a helix as defined above is $\frac{|r|}{r^2+p^2}$ and its torsion is $\frac{p}{r^2+p^2}.$