I'd start with a long soak in vinegar and water to disolve residue from hard water, then scrub gently with a soft toothbrush.

Here's an article on cleaning whirlpool tubs. How to clean a jetted tub

You should also download the manual for your tub and check the care instructions. Here's what one manual for Kohler says, in part:

Flush your whirlpool system twice a month or more, depending upon usage. Use the following sequence:

1. Adjust the jets so there is no air induction;
2. Fill the whirlpool with hot water to a level above the back jet water sensors;
3. Add 2 teaspoons of a low-foaming dishwasher detergent, and 1/2 cup of household bleach to the water;
4. Run the whirlpool for 5 to 10 minutes in each massage mode; then drain;
5. Fill with cold water to a level above the back jet water-level sensors, and run the whirlpool for another 5 to 10 minutes in each massage mode.
6. Drain the whirlpool, and wipe it dry with a soft cloth.