1. **State the problem:** We need to find the cosecant of angle $K$ in the right triangle $KIJ$ where $\angle I$ is the right angle. 2. **Identify the sides:** Given side lengths are $KI=12$, $IJ=35$, and hypotenuse $KJ=37$. 3. **Recall the definition:** Cosecant is the reciprocal of sine. For angle $K$, $\sin(K) = \frac{\text{opposite}}{\text{hypotenuse}}$. 4. **Determine opposite side to $K$:** Opposite side to $K$ is $IJ=35$. 5. **Calculate $\sin(K)$:** $$\sin(K) = \frac{35}{37}$$ 6. **Calculate $\csc(K)$:** $$\csc(K) = \frac{1}{\sin(K)} = \frac{1}{\frac{35}{37}} = \frac{37}{35}$$ 7. **Simplify if possible:** $\frac{37}{35}$ is already in simplest form. **Final answer:** $$\boxed{\csc(K) = \frac{37}{35}}$$