Step 1: Convert the given quantities into a mass balance:
Distilled water has a density of $1.0$ g/cm$^3$, which numerically means every 1 ml of water weighs exactly 1 gram. So 92 ml of water corresponds directly to 92 grams of water, without needing any further unit conversion.
Step 2: Add the mass of the dissolved salt:
When 23 grams of NaCl salt dissolves into the water, the total mass of the system is simply the sum of both components, since mass is conserved during dissolution: $m_{total} = 92 \text{ g} + 23 \text{ g} = 115 \text{ g}$.
Step 3: Use the given brine density to back calculate volume:
The problem states the resultant brine has a density of $1.15$ g/cm$^3$. Density is mass per unit volume, so rearranging gives $V_{brine} = \dfrac{m_{total}}{\rho_{brine}}$.
Step 4: Substitute the numbers and solve:
$V_{brine} = \dfrac{115}{1.15} = 100$ cm$^3$. Converting to milliliters using the standard equivalence $1 \text{ cm}^3 = 1 \text{ ml}$ gives $V_{brine} = 100$ ml. This is a good example of why volumes are not simply additive when a solid dissolves into a liquid to form a solution of different density; only the masses add directly, while the final volume must be recalculated from the new density.
Final Answer:
\[\boxed{100.0 \text{ ml}}\]