What is the magnitude of the tangential, normal and total accelerations here

Your derivative is wrong. If you take the time derivative of $vec{v}$, you should get $vec{a}=frac{dvec{v}}{dt}=bhat{j}$.

Note that in this case, it's not correct to take the time derivative of the magnitude $|vec{v}|$. This only tells you the rate of change of speed, but this is not equal to $|vec{a}|$.

Consider the case where a point is moving around a circle at constant speed. In this case, we have $frac{d|vec{v}|}{dt}=0$, since the speed is a constant. However, $aneq 0$. This is because the direction of $vec{v}$ is changing, and some acceleration is needed to make this change happen.

Note that this equation resembles the one for a projectile coming down (with +y down). If you put the given time in the velocity equation, you get the components (and direction) for the velocity at that time. The acceleration is in the y direction, but you can get the components of the acceleration which are parallel and perpendicular to the velocity.