There is a complex number $K$ such that the quadratic polynomial $7x^2 +Kx + 12 - 5i$ has exactly one root, where $i =\sqrt{-1}$. Find $|K|^2$.
1. **Determine the condition for a double root:**
   For the quadratic polynomial \(7x^2 + Kx + (12 - 5i)\) to have exactly one root, the discriminant must be zero. The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   Here, \(a = 7\), \(b = K\), and \(c = 12 - 5i\). Therefore, the discriminant is:
   \[
   \Delta = K^2 - 4 \cdot 7 \cdot (12 - 5i)
   \]

2. **Set the discriminant to zero:**
   \[
   K^2 - 28(12 - 5i) = 0
   \]
   Simplifying inside the parentheses:
   \[
   K^2 - 336 + 140i = 0
   \]
   Therefore:
   \[
   K^2 = 336 - 140i
   \]

3. **Express \(K\) in terms of real and imaginary parts:**
   Let \(K = a + bi\), where \(a\) and \(b\) are real numbers. Then:
   \[
   K^2 = (a + bi)^2 = a^2 - b^2 + 2abi
   \]
   Equate the real and imaginary parts:
   \[
   a^2 - b^2 = 336 \quad \text{(1)}
   \]
   \[
   2ab = -140 \quad \text{(2)}
   \]

4. **Solve for \(a\) and \(b\):**
   From equation (2):
   \[
   ab = -70
   \]
   Substitute \(b = -\frac{70}{a}\) into equation (1):
   \[
   a^2 - \left(-\frac{70}{a}\right)^2 = 336
   \]
   Simplify:
   \[
   a^2 - \frac{4900}{a^2} = 336
   \]
   Multiply through by \(a^2\) to clear the fraction:
   \[
   a^4 - 336a^2 - 4900 = 0
   \]
   Let \(x = a^2\). Then the equation becomes:
   \[
   x^2 - 336x - 4900 = 0
   \]

5. **Solve the quadratic equation for \(x\):**
   Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
   \[
   x = \frac{336 \pm \sqrt{336^2 + 4 \cdot 4900}}{2}
   \]
   \[
   x = \frac{336 \pm \sqrt{112896 + 19600}}{2}
   \]
   \[
   x = \frac{336 \pm \sqrt{132496}}{2}
   \]
   \[
   x = \frac{336 \pm 364}{2}
   \]
   This gives two solutions:
   \[
   x = \frac{336 + 364}{2} = 350 \quad \text{and} \quad x = \frac{336 - 364}{2} = -14
   \]
   Since \(x = a^2\) must be non-negative, we have:
   \[
   a^2 = 350
   \]
   Therefore:
   \[
   b^2 = \frac{4900}{a^2} = \frac{4900}{350} = 14
   \]

6. **Find \(|K|^2\):**
   \[
   |K|^2 = a^2 + b^2 = 350 + 14 = 364
   \]

The final answer is \(\boxed{364}\).