The unlimited sick leave policy is meant to apply to those who are sick. If someone is sick, you can react in two ways.
Take the person at their word that they are sick - This reaction is most appropriate when the person is sick for a short period of time. If I am rolling on the floor because I ate something that I should have thrown out six months ago, don't make me drag myself to the doctor's office :)
Make the person substantiate their claim that they are sick - If the employee can make it to the nightclub after hours, surely he can make it to the doctor's office. In fact, if he is taking that many sick days, he SHOULD be going to the doctor's office.
You should ask the employee if he can work from home - No employee who operates in good faith would turn down working from home if working from home helps him get around his medical condition.