1. **State the problem:** Solve for $k$ in the equation $$9 - \frac{59}{60} = -\frac{6}{5} + \frac{1}{5}k.$$\n\n2. **Rewrite the equation:** \n$$9 - \frac{59}{60} = -\frac{6}{5} + \frac{1}{5}k.$$\n\n3. **Convert 9 to a fraction with denominator 60:** \n$$9 = \frac{540}{60}.$$\n\n4. **Subtract the fractions on the left side:** \n$$\frac{540}{60} - \frac{59}{60} = \frac{540 - 59}{60} = \frac{481}{60}.$$\n\n5. **Rewrite the equation:** \n$$\frac{481}{60} = -\frac{6}{5} + \frac{1}{5}k.$$\n\n6. **Add $\frac{6}{5}$ to both sides:** \n$$\frac{481}{60} + \frac{6}{5} = \frac{1}{5}k.$$\n\n7. **Convert $\frac{6}{5}$ to denominator 60:** \n$$\frac{6}{5} = \frac{72}{60}.$$\n\n8. **Add the fractions:** \n$$\frac{481}{60} + \frac{72}{60} = \frac{553}{60}.$$\n\n9. **Rewrite:** \n$$\frac{553}{60} = \frac{1}{5}k.$$\n\n10. **Multiply both sides by 5 to solve for $k$:** \n$$5 \times \frac{553}{60} = k.$$\n\n11. **Simplify:** \n$$k = \frac{5 \times 553}{60} = \frac{2765}{60}.$$\n\n12. **Simplify the fraction by dividing numerator and denominator by 5:** \n$$k = \frac{\cancel{5} \times 553}{\cancel{5} \times 12} = \frac{553}{12}.$$\n\n13. **Final answer:** \n$$k = \frac{553}{12}.$$