Finding equilibrium of perfect competition in the short run with a cost function
In these problems, you are generally dealing with identical firms - all of whom will supply the market according to their marginal cost curve.  If we were solving for a long run equilibrium the first thing we would do is get the supply curve - which can be done by finding the optimal mix of capital and labor via:

MPL/W = MPK/r

This will get you L = 4K.

In this situation you can skip to:
$$
Q = \sqrt{3}\sqrt{L}\\
L = Q^{2}/3
$$

Plug those into $C(Q) = Kr + Lw$

$$
C(Q) = 108 + 3Q^{2}\,
$$

The fixed cost is \$108 (but that is sunk in the short run, so the firm will ignore it), the variable cost is $3Q^{2}$, making the marginal cost $6Q$.  For the market, this will be $\frac{Q}{1.5}$ (the horizontal sum of all firms' production)

Marginal revenue in this market is just the price since we assume it is perfect competition, so setting market supply = demand gives you
$$
360 - 2Q = \frac{Q}{1.5}\\
Q = 135\,
$$

$Q/n = q$  tells us that each of the 9 firms must produce 15 units for the market to supply 135 total units.  According to the demand curve, the market price is \$90 when 135 units are sold.

Total surplus can be found by calculating the area of two triangles:

$$
CS = (1/2)135*(360-90) = \$18,225\\
PS = (1/2)135*(90) = \$6,075\\
TS = \$24,300\,
$$