Calculation of a current in the Laplace domain
Your first equation is wrong, or I don't grasp your notation. I equals the applied voltage divided by the impedance, so the first formula you use is not quite right. The third term if different from the second, and should make zero since you are summing all the voltages along a closed loop.

The proper way to tackle this problem is apply ohm's law:

$$
V = I\cdot R \rightarrow I = \frac{V}{R}
$$

The law is still valid in the s domain if you use the impedances:

$$
I(s) = \frac{V(s)}{Z(s)}
$$

In your circuit you have an impedance that is the series of a capacitor and an inductor, thus:
$$
Z(s) = sL + \frac{1}{Cs}
$$

While \$V(s)\$ is given. Finally:
$$
I(s) = \frac{\frac{V_0}{s}}{sL + \frac{1}{Cs}}
$$

You can have a little more fun and come to:
$$
I(s) = V_0\frac{C}{s^2LC+1}
$$