A

To determine the height from which the ice should fall to melt completely, we need to consider the energy transformations involved.

Step 1: Calculate the energy required to melt the ice.
The latent heat of fusion for ice is given as \( L = 3.4 \times 10^5 \: \text{J/kg} \). If we let the mass of the ice be \( m \) kg, the total energy required to melt the ice completely can be expressed as:
\[ Q = mL = m(3.4 \times 10^5) \: \text{J} \]

Step 2: Calculate the gravitational potential energy gained by the ice when it falls from a height \( h \).
The potential energy (PE) at height \( h \) is given by:
\[ PE = mgh = mg(10)h = 10mh \]

Step 3: Set up the equation considering that only one-quarter of the energy produced is absorbed by ice:
\[ \frac{1}{4}PE = Q \]
Thus, we have:
\[ \frac{1}{4}(10mh) = m(3.4 \times 10^5) \]

Step 4: Simplify the equation:
The mass \( m \) cancels out (providing \( m \neq 0 \)):
\[ \frac{10h}{4} = 3.4 \times 10^5 \]

Step 5: Solve for height \( h \):
\[ 10h = 4 \times 3.4 \times 10^5 \]
\[ h = \frac{4 \times 3.4 \times 10^5}{10} \]
\[ h = \frac{13.6 \times 10^5}{10} = 1.36 \times 10^5 \: ext{m} = 136000 ext{ m} \]

Therefore, the height from which the ice should fall to melt completely is 136,000 m (or 136 km). Hence, the correct answer is option A.